LOCAL CONSERVATION LAWS AND THE HAMILTONIAN FORMALISM FOR THE ABLOWITZ–LADIK HIERARCHY

FRITZ GESZTESY, HELGE HOLDEN, JOHANNA MICHOR, AND GERALD TESCHL

Abstract. We derive a systematic and recursive approach to local conservation laws and the Hamiltonian formalism for the Ablowitz–Ladik (AL) hierarchy. Our methods rely on a recursive approach to the AL hierarchy using Laurent polynomials and on asymptotic expansions of the Green’s function of the AL Lax operator, a five-diagonal finite difference operator.

1. Introduction

The principal aim of this paper is to provide a systematic and recursive approach to local conservation laws and the Hamiltonian formalism for the Ablowitz–Ladik (AL) hierarchy of integrable differential-difference equations.

Consider sequences {α(n, t), β(n, t)}_{n∈ℤ} ∈ ℓ^{1}(ℤ) satisfying some additional assumptions to be specified later, parametrized by the deformation (time) parameter t ∈ ℝ, that are solutions of the Ablowitz–Ladik equations

(\begin{matrix}−iα_{t} − (1 − αβ)(α^{−} + α^{+}) + 2α \\ −iβ_{t} + (1 − αβ)(β^{−} + β^{+}) − 2β \end{matrix}) = 0.  (1.1)

Here c^{±} denote shifts, that is, c^{±}(n) = c(n ± 1), n ∈ ℤ. Then clearly

∂_{t} ∑_{n∈ℤ} α^{+}(n, t)β(n, t) = ∂_{t} ∑_{n∈ℤ} α(n, t)β^{+}(n, t) = 0.  (1.2)

Indeed, one can show the existence of an infinite sequence {ρ_{j,±}}_{j∈ℕ} of polynomials of α, β and certain shifts thereof, with the property that the lattice sum is timeindependent,

∂_{t} ∑_{n∈ℤ} ρ_{j,±}(n, t) = 0, j ∈ ℕ.  (1.3)

This result is obtained by deriving local conservation laws of the type

∂_{t}ρ_{j,±} + (S^{+} − I)J_{j,±} = 0, j ∈ ℕ,  (1.4)

for certain polynomials J_{j,±} of α, β and certain shifts thereof. The polynomials J_{j,±} will be constructed via an explicit recursion relation. For a detailed discussion of these results we refer to Theorem 5.7 and Remarks 5.8 and refrALh5.8a.

Date: October 29, 2007.

2000 Mathematics Subject Classification. Primary 37K10, 37K20, 47B36; Secondary 35Q58, 37K60.

Key words and phrases. Ablowitz–Ladik hierarchy, Hamiltonian formalism, conservation laws.

Research supported in part by the Research Council of Norway, the US National Science Foundation under Grant No. DMS-0405526, and the Austrian Science Fund (FWF) under Grants No. Y330, J2655.

The above analysis extends to the full Ablowitz–Ladik hierarchy as follows. The \underline{p}th equation, \underline{p} = (p_{−}, p_{+}) ∈ ℕ_{0}^{2} (where ℕ_{0} = ℕ∪\{0\}), in the AL hierarchy is given by

\begin{matrix}AL\underline{p}(α, β) = (\begin{matrix}−iα_{t\underline{p}} − α(g_{p_{+},+} + g_{p_{−},−}^{−}) + f_{p_{+}−1,+} − f_{p_{−}−1,−}^{−} \\ −iβ_{t\underline{p}} + β(g_{p_{+},+}^{−} + g_{p_{−},−}) − h_{p_{−}−1,−} + h_{p_{+}−1,+}^{−} \end{matrix}) = 0, \\ t\underline{p} ∈ ℝ, \underline{p} = (p_{−}, p_{+}) ∈ ℕ_{0}^{2}, \end{matrix}  (1.5)

where f_{ℓ,±}, g_{ℓ,±}, and h_{ℓ,±} are carefully designed polynomial expressions of α, β and certain shifts thereof. Recursively, they are given by (2.5)–(2.12). On each level in the recursion an arbitrary constant c_{ℓ,±} ∈ ℂ is introduced. In the homogeneous case, where all these constants c_{ℓ}, ℓ ∈ ℕ, are set equal to zero, a hat ˆ is added in the notation, that is, f̂_{ℓ,±}, ĝ_{ℓ,±}, ĥ_{ℓ,±}, etc., denote the corresponding homogeneous quantities. The homogeneous coefficients f̂_{ℓ,±}, ĝ_{ℓ,±}, ĥ_{ℓ,±} can also be expressed explicitly in terms of appropriate matrix elements of powers of the AL Lax finite difference expression L defined in (3.3), (3.5) and the finite difference expressions D and E in (3.14), as described in Lemma 3.1. The conserved densities ρ_{j,±} are independent of the equation in the hierarchy while the currents J_{\underline{p},j,±} depend on \underline{p}; thus one finds (cf. Theorem 5.7)

∂_{t\underline{p}}ρ_{j,±} + (S^{+} − I)J_{\underline{p},j,±} = 0, t\underline{p} ∈ ℝ, j ∈ ℕ, \underline{p} ∈ ℕ_{0}^{2}.  (1.6)

For α, β ∈ ℓ^{1}(ℤ) it then follows that

\frac{d}{dt\underline{p}} ∑_{n∈ℤ} ρ_{j,±}(n, t\underline{p}) = 0, t\underline{p} ∈ ℝ, j ∈ ℕ, \underline{p} ∈ ℕ_{0}^{2}.  (1.7)

By showing that ρ_{j,±} equals ĝ_{j,±} up to a first-order difference expression (cf. Lemma 4.4), and by investigating the time-dependence of γ = 1 − αβ, one concludes (cf. Remark 5.8) that

\frac{d}{dt\underline{p}} ∑_{n∈ℤ} ln(γ(n, t\underline{p})) = 0, \frac{d}{dt\underline{p}} ∑_{n∈ℤ} ĝ_{j,±}(n, t\underline{p}) = 0, t\underline{p} ∈ ℝ, j ∈ ℕ, \underline{p} ∈ ℕ_{0}^{2}, (1.8)

represent the two infinite sequences of AL conservation laws. Our approach to (1.6) is based on a careful analysis of asymptotic expansions of the Green’s function (as the spectral parameter tends to zero and to infinity) for the operator realization L̆ in ℓ^{2}(ℤ) corresponding to the Lax difference expression L in (3.3), (3.5).

In addition, we provide a detailed study of the Hamiltonian formalism for the AL hierarchy. In particular, the \underline{p}th equation in the AL hierarchy can be written as (cf. Theorem 6.5)

AL\underline{p}(α, β) = (\begin{matrix}−iα_{t\underline{p}} \\ −iβ_{t\underline{p}} \end{matrix}) + D∇H\underline{p} = 0, \underline{p} ∈ ℕ_{0}^{2},  (1.9)

where the Hamiltonians H\underline{p} are given by

H\underline{p} = ∑_{ℓ=1}^{p_{+}} c_{p_{+}−ℓ,+}Ĥ_{ℓ,+} + ∑_{ℓ=1}^{p_{−}} c_{p_{−}−ℓ,−}Ĥ_{ℓ,−} + c\underline{p}Ĥ_{0}, \underline{p} = (p_{−}, p_{+}) ∈ ℕ_{0}^{2}, (1.10)

Ĥ_{0} = ∑_{n∈ℤ} ln(γ(n)),  Ĥ_{p_{±},±} = \frac{1}{p_{±}} ∑_{n∈ℤ} ĝ_{p_{±},±}(n), p_{±} ∈ ℕ.  (1.11)

Here D = (1 − αβ) ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ). Furthermore, we show that any H\underline{r} is conserved by the Hamiltonian flows in (1.9) (cf. Theorem 6.6), that is,

\frac{dH\underline{r}}{dt\underline{p}} = 0, \underline{p}, \underline{r} ∈ ℕ_{0}^{2}.  (1.12)

Moreover, for general sequences α, β (i.e., not assuming that they satisfy an equation in the AL hierarchy), we show in Theorem 6.7 that

\{Ĥ\underline{p}, Ĥ\underline{r}\} = 0, \underline{p}, \underline{r} ∈ ℕ_{0}^{2},  (1.13)

for a suitably defined Poisson bracket \{ · , · \} (see (6.16)), that is, H\underline{p} and H\underline{r} are in involution for all \underline{p}, \underline{r} ∈ ℕ_{0}^{2}.

The Ablowitz–Ladik hierarchy has been extensively discussed in the completely integrable system literature (cf., e.g., [3]–[6], [1], [2, Sect. 3.2.2], [7, Ch. 3], [12], [13], [34], [36], [37], [38], [41], [45], [47], [48] and the references cited therein) and in recent years especially due to its close connections with the theory of orthogonal polynomials, a field that underwent a remarkable resurgency in recent years (cf. [42], [43], [44] and the quoted therein). Rather than repeating some of the AL hierarchy history and its relevance to the theory of orthogonal polynomials at this place, we refer to the detailed introductions of [25], [26], [27] and the extensive bibliography listed therein. Here we just mention references intimately connected with the topics discussed in this paper: Infinitely many conservation laws are discussed, for instance, by Ablowitz and Ladik [4], Ablowitz, Prinari, and Trubatch [7, Ch. 3], Ding, Sun, and Xu [14], Zhang and Chen [50], and Zhang, Ning, Bi, and Chen [52]; the bi-Hamiltonian structure of the AL hierarchy is considered by Ercolani and Lozano [15], Hydon [30], and Lozano [33], multi-Hamiltonian structures for the defocusing AL hierarchy were studied by Gekhtman and Nenciu [18], Zeng and Rauch-Wojciechowski [49], and Zhang and Chen [51]; Poisson brackets for orthogonal polynomials on the unit circle relevant to the case of the defocusing AL hierarchy (where β = \overline{α}) have been studied by Cantero and Simon [10], Killip and Nenciu [31], and Nenciu [39]; Lenard recursions and Hamiltonian structures were discussed in Geng and Dai [19] and Geng, Dai, and Zhu [20].

Next we briefly describe the structure of this paper: Section 2 recalls the recursive construction of the AL hierarchy as discussed in detail in [25] (see also [26], [27]). In Section 3 we introduce the Lax pair for the AL hierarchy and prove its equivalence with the corresponding zero-curvature formulation. These results are new. In Section 4 we discuss the Green’s function of the Lax operator L̆ and study its asymptotic expansions as the spectral parameter tends to zero and infinity. As a direct consequence of these asymptotic expansions, local conservation laws are then derived in Section 5. Our final Section 6 then introduces the basics of variational derivatives and provides a detailed derivation of the Hamiltonian formalism for the AL hierarchy.

Finally, we emphasize that our recursive and systematic approach to local conservation laws of the Ablowitz–Ladik hierarchy appears to be new. Moreover, our treatment of Poisson brackets and variational derivatives, and their connections with the diagonal Green’s function of the underlying Lax operator, now puts the AL hierarchy on precisely the same level as the Toda and KdV hierarchy with respect to this particular aspect of the Hamiltonian formalism (cf. [22, Ch. 1], [23, Ch. 1]).

2. The Ablowitz–Ladik hierarchy in a nutshell

In this section we summarize the construction of the Ablowitz–Ladik hierarchy employing a Laurent polynomial recursion formalism and derive the associated sequence of Ablowitz–Ladik zero-curvature pairs. Moreover, we discuss the Burchnall–Chaundy Laurent polynomial in connection with the stationary Ablowitz–Ladik hierarchy and the underlying hyperelliptic curve. For a detailed treatment of this material we refer to [23], [25].

We denote by ℂ^{ℤ} the set of complex-valued sequences indexed by ℤ.

Throughout this section we suppose the following hypothesis.

Hypothesis 2.1. In the stationary case we assume that α, β satisfy

α, β ∈ ℂ^{ℤ}, α(n)β(n) ∉ {0, 1}, n ∈ ℤ.  (2.1)

In the time-dependent case we assume that α, β satisfy

\begin{matrix}α( · , t), β( · , t) ∈ ℂ^{ℤ}, t ∈ ℝ, α(n, ·), β(n, ·) ∈ C^{1}(ℝ), n ∈ ℤ, \\ α(n, t)β(n, t) ∉ \{0, 1\}, (n, t) ∈ ℤ × ℝ. \end{matrix}  (2.2)

We denote by S^{±} the shift operators acting on complex-valued sequences f = {f(n)}_{n∈ℤ} ∈ ℂ^{ℤ} according to

(S^{±}f)(n) = f(n ± 1), n ∈ ℤ.  (2.3)

Moreover, we will frequently use the notation

f^{±} = S^{±}f, f ∈ ℂ^{ℤ}.  (2.4)

To construct the Ablowitz–Ladik hierarchy one typically introduces appropriate zero-curvature pairs of 2 × 2 matrices, denoted by U(z) and V\underline{p}(z), \underline{p} ∈ ℕ_{0}^{2} (with z ∈ ℂ \ {0} a certain spectral parameter to be discussed later), defined recursively in the following. We take the shortest route to the construction of V\underline{p} and hence to that of the Ablowitz–Ladik hierarchy by starting from the recursion relation (2.5)–(2.12) below.

Define sequences {f_{ℓ,±}}_{ℓ∈ℕ_{0}}, {g_{ℓ,±}}_{ℓ∈ℕ_{0}}, and {h_{ℓ,±}}_{ℓ∈ℕ_{0}} recursively by

g_{0,+} = \frac{1}{2}c_{0,+}, f_{0,+} = −c_{0,+}α^{+}, h_{0,+} = c_{0,+}β,  (2.5)
g_{ℓ+1,+} − g_{ℓ+1,+}^{−} = αh_{ℓ,+}^{−} + βf_{ℓ,+}, ℓ ∈ ℕ_{0},  (2.6)
f_{ℓ+1,+}^{−} = f_{ℓ,+} − α(g_{ℓ+1,+} + g_{ℓ+1,+}^{−}), ℓ ∈ ℕ_{0},  (2.7)
h_{ℓ+1,+} = h_{ℓ,+}^{−} + β(g_{ℓ+1,+} + g_{ℓ+1,+}^{−}), ℓ ∈ ℕ_{0},  (2.8)

and

g_{0,−} = \frac{1}{2}c_{0,−}, f_{0,−} = c_{0,−}α, h_{0,−} = −c_{0,−}β^{+},  (2.9)
g_{ℓ+1,−} − g_{ℓ+1,−}^{−} = αh_{ℓ,−} + βf_{ℓ,−}^{−}, ℓ ∈ ℕ_{0},  (2.10)
f_{ℓ+1,−} = f_{ℓ,−}^{−} + α(g_{ℓ+1,−} + g_{ℓ+1,−}^{−}), ℓ ∈ ℕ_{0},  (2.11)
h_{ℓ+1,−}^{−} = h_{ℓ,−} − β(g_{ℓ+1,−} + g_{ℓ+1,−}^{−}), ℓ ∈ ℕ_{0}.  (2.12)

Here c_{0,±} ∈ ℂ are given constants. For later use we also introduce

f_{−1,±} = h_{−1,±} = 0.  (2.13)

Remark 2.2. The sequences {f_{ℓ,+}}_{ℓ∈ℕ_{0}}, {g_{ℓ,+}}_{ℓ∈ℕ_{0}}, and {h_{ℓ,+}}_{ℓ∈ℕ_{0}} can be computed recursively as follows: Assume that f_{ℓ,+}, g_{ℓ,+}, and h_{ℓ,+} are known. Equation (2.6) is a first-order difference equation in g_{ℓ+1,+} that can be solved directly and yields a local lattice function that is determined up to a new constant denoted by c_{ℓ+1,+} ∈ ℂ. Relations (2.7) and (2.8) then determine f_{ℓ+1,+} and h_{ℓ+1,+}, etc. The sequences {f_{ℓ,−}}_{ℓ∈ℕ_{0}}, {g_{ℓ,−}}_{ℓ∈ℕ_{0}}, and {h_{ℓ,−}}_{ℓ∈ℕ_{0}} are determined similarly.

Upon setting

γ = 1 − αβ,  (2.14)

one explicitly obtains

\begin{aligned}f_{0,+} & = c_{0,+}(−α^{+}), f_{1,+} = c_{0,+}( − γ^{+}α^{++} + (α^{+})^{2}β) + c_{1,+}(−α^{+}), \\ g_{0,+} & = \frac{1}{2}c_{0,+}, g_{1,+} = c_{0,+}(−α^{+}β) + \frac{1}{2}c_{1,+}, \\ h_{0,+} & = c_{0,+}β, h_{1,+} = c_{0,+}(γβ^{−} − α^{+}β^{2}) + c_{1,+}β, \\ f_{0,−} & = c_{0,−}α, f_{1,−} = c_{0,−}(γα^{−} − α^{2}β^{+}) + c_{1,−}α, \\ g_{0,−} & = \frac{1}{2}c_{0,−}, g_{1,−} = c_{0,−}(−αβ^{+}) + \frac{1}{2}c_{1,−}, \\ h_{0,−} & = c_{0,−}(−β^{+}), h_{1,−} = c_{0,−}( − γ^{+}β^{++} + α(β^{+})^{2}) + c_{1,−}(−β^{+}), etc. \end{aligned} (2.15)

Here {c_{ℓ,±}}_{ℓ∈ℕ} denote summation constants which naturally arise when solving the difference equations for g_{ℓ,±} in (2.6), (2.10). Subsequently, it will also be useful to work with the corresponding homogeneous coefficients f̂_{ℓ,±}, ĝ_{ℓ,±}, and ĥ_{ℓ,±}, defined by the vanishing of all summation constants c_{k,±} for k = 1, …, ℓ, and choosing c_{0,±} = 1,

f̂_{0,+} = −α^{+}, f̂_{0,−} = α, f̂_{ℓ,±} = f_{ℓ,±}|_{c_{0,±}=1, c_{j,±}=0,j=1,...,ℓ}, ℓ ∈ ℕ,  (2.16)
ĝ_{0,±} = \frac{1}{2}, ĝ_{ℓ,±} = g_{ℓ,±}|_{c_{0,±}=1, c_{j,±}=0,j=1,...,ℓ}, ℓ ∈ ℕ,  (2.17)
ĥ_{0,+} = β, ĥ_{0,−} = −β^{+}, ĥ_{ℓ,±} = h_{ℓ,±}|_{c_{0,±}=1, c_{j,±}=0,j=1,...,ℓ}, ℓ ∈ ℕ. (2.18)

By induction one infers that

f_{ℓ,±} = ∑_{k=0}^{ℓ} c_{ℓ−k,±}f̂_{k,±}, g_{ℓ,±} = ∑_{k=0}^{ℓ} c_{ℓ−k,±}ĝ_{k,±}, h_{ℓ,±} = ∑_{k=0}^{ℓ} c_{ℓ−k,±}ĥ_{k,±}. (2.19)

In a slight abuse of notation we will occasionally stress the dependence of f_{ℓ,±}, g_{ℓ,±}, and h_{ℓ,±} on α, β by writing f_{ℓ,±}(α, β), g_{ℓ,±}(α, β), and h_{ℓ,±}(α, β).

One can show (cf. [25]) that all homogeneous elements f̂_{ℓ,±}, ĝ_{ℓ,±}, and ĥ_{ℓ,±}, ℓ ∈ ℕ_{0}, are polynomials in α, β, and some of their shifts.

Remark 2.3. As an efficient tool to distinguish between nonhomogeneous and homogeneous quantities f_{ℓ,±}, g_{ℓ,±}, h_{ℓ,±}, and f̂_{ℓ,±}, ĝ_{ℓ,±}, ĥ_{ℓ,±}, respectively, we now introduce the notion of degree as follows. Denote

f^{(r)} = S^{(r)}f, f = \{f(n)\}_{n∈ℤ} ∈ ℂ^{ℤ}, S^{(r)} = \begin{cases}(S^{+})^{r}, & r ≥ 0, \\ (S^{−})^{−r}, & r < 0, \end{cases} r ∈ ℤ, (2.20)

and define

deg (α^{(r)}) = r, deg (β^{(r)}) = −r, r ∈ ℤ.  (2.21)

This implies

deg (f̂_{ℓ,+}^{(r)}) = ℓ + 1 + r, deg (f̂_{ℓ,−}^{(r)}) = −ℓ + r, deg (ĝ_{ℓ,±}^{(r)}) = ±ℓ,
deg (ĥ_{ℓ,+}^{(r)}) = ℓ − r, deg (ĥ_{ℓ,−}^{(r)}) = −ℓ − 1 − r, ℓ ∈ ℕ_{0}, r ∈ ℤ.

(2.22)

Alternatively the homogeneous coefficients can be computed directly via the following nonlinear recursion relations:

Lemma 2.4. The homogeneous quantities f̂_{ℓ,±}, ĝ_{ℓ,±}, ĥ_{ℓ,±} are uniquely defined by the following recursion relations:

\begin{aligned}ĝ_{0,+} & = \frac{1}{2}, f̂_{0,+} = −α^{+}, ĥ_{0,+} = β, \\ ĝ_{l+1,+} & = ∑_{k=0}^{l} f̂_{l−k,+}ĥ_{k,+} − ∑_{k=1}^{l} ĝ_{l+1−k,+}ĝ_{k,+}, \\ f̂_{l+1,+}^{−} & = f̂_{l,+} − α(ĝ_{l+1,+} + ĝ_{l+1,+}^{−}), \\ ĥ_{l+1,+} & = ĥ_{l,+}^{−} + β(ĝ_{l+1,+} + ĝ_{l+1,+}^{−}), \end{aligned}  (2.23)

and

\begin{aligned}ĝ_{0,−} & = \frac{1}{2}, f̂_{0,−} = α, ĥ_{0,−} = −β^{+}, \\ ĝ_{l+1,−} & = ∑_{k=0}^{l} f̂_{l−k,−}ĥ_{k,−} − ∑_{k=1}^{l} ĝ_{l+1−k,−}ĝ_{k,−}, \\ f̂_{l+1,−} & = f̂_{l,−}^{−} + α(ĝ_{l+1,−} + ĝ_{l+1,−}^{−}), \\ ĥ_{l+1,−}^{−} & = ĥ_{l,−} − β(ĝ_{l+1,−} + ĝ_{l+1,−}^{−}). \end{aligned}  (2.24)

We also note the following useful result (cf. [25]): Assuming (2.1), we find

\begin{aligned}g_{ℓ,+} − g_{ℓ,+}^{−} & = αh_{ℓ,+} + βf_{ℓ,+}^{−}, ℓ ∈ ℕ_{0}, \\ g_{ℓ,−} − g_{ℓ,−}^{−} & = αh_{ℓ,−}^{−} + βf_{ℓ,−}, ℓ ∈ ℕ_{0}. \end{aligned}  (2.25)

Moreover, we record the following symmetries,

f̂_{ℓ,±}(c_{0,±}, α, β) = ĥ_{ℓ,∓}(c_{0,∓}, β, α), ĝ_{ℓ,±}(c_{0,±}, α, β) = ĝ_{ℓ,∓}(c_{0,∓}, β, α), ℓ ∈ ℕ_{0}.

(2.26)

Next we define the 2 × 2 zero-curvature matrices

U(z) = (\begin{matrix}z & α \\ zβ & 1 \end{matrix})  (2.27)

and

V\underline{p}(z) = i (\begin{matrix}G− ; \underline{p}(z) & −F − ; \underline{p}(z) \\ H− ; \underline{p}(z) & −K− ; \underline{p}(z) \end{matrix}) , \underline{p} ∈ ℕ_{0}^{2},  (2.28)

for appropriate Laurent polynomials F\underline{p}(z), G\underline{p}(z), H\underline{p}(z), and K\underline{p}(z) in the spectral parameter z ∈ ℂ\{0} to be determined shortly. By postulating the stationary zerocurvature relation,

0 = UV\underline{p} − V + ; \underline{p}U,  (2.29)

one concludes that (2.29) is equivalent to the following relations

z(G− ; \underline{p} − G\underline{p}) + zβF\underline{p} + αH− ; \underline{p} = 0,  (2.30)

zβF − ; \underline{p} + αH\underline{p} − K\underline{p} + K− ; \underline{p} = 0,  (2.31)
−F\underline{p} + zF − ; \underline{p} + α(G\underline{p} + K− ; \underline{p}) = 0,  (2.32)
zβ(G− ; \underline{p} + K\underline{p}) − zH\underline{p} + H− ; \underline{p} = 0.  (2.33)

In order to make the connection between the zero-curvature formalism and the recursion relations (2.5)–(2.12), we now define Laurent polynomials F\underline{p}, G\underline{p}, H\underline{p}, and K\underline{p}, \underline{p} = (p_{−}, p_{+}) ∈ ℕ_{0}^{2}, by^{1}

F\underline{p}(z) = ∑_{ℓ=1}^{p_{−}} f_{p_{−}−ℓ,−}z^{−ℓ} + ∑_{ℓ=0}^{p_{+}−1} f_{p_{+}−1−ℓ,+}z^{ℓ},  (2.34)

G\underline{p}(z) = ∑_{ℓ=1}^{p_{−}} g_{p_{−}−ℓ,−}z^{−ℓ} + ∑_{ℓ=0}^{p_{+}} g_{p_{+}−ℓ,+}z^{ℓ},  (2.35)

H\underline{p}(z) = ∑_{ℓ=0}^{p_{−}−1} h_{p_{−}−1−ℓ,−}z^{−ℓ} + ∑_{ℓ=1}^{p_{+}} h_{p_{+}−ℓ,+}z^{ℓ},  (2.36)

K\underline{p}(z) = ∑_{ℓ=0}^{p_{−}} g_{p_{−}−ℓ,−}z^{−ℓ} + ∑_{ℓ=1}^{p_{+}} g_{p_{+}−ℓ,+}z^{ℓ} = G\underline{p}(z) + g_{p_{−},−} − g_{p_{+},+}.  (2.37)

The corresponding homogeneous quantities are defined by (ℓ ∈ ℕ_{0})

\begin{aligned}F̂_{0,∓}(z) & = 0, F̂_{ℓ,−}(z) = ∑_{k=1}^{ℓ} f̂_{ℓ−k,−}z^{−k}, F̂_{ℓ,+}(z) = ∑_{k=0}^{ℓ−1} f̂_{ℓ−1−k,+}z^{k}, \\ Ĝ_{0,−}(z) & = 0, Ĝ_{ℓ,−}(z) = ∑_{k=1}^{ℓ} ĝ_{ℓ−k,−}z^{−k}, \\ Ĝ_{0,+}(z) & = \frac{1}{2}, Ĝ_{ℓ,+}(z) = ∑_{k=0}^{ℓ} ĝ_{ℓ−k,+}z^{k}, \\ Ĥ_{0,∓}(z) & = 0, Ĥ_{ℓ,−}(z) = ∑_{k=0}^{ℓ−1} ĥ_{ℓ−1−k,−}z^{−k}, Ĥ_{ℓ,+}(z) = ∑_{k=1}^{ℓ} ĥ_{ℓ−k,+}z^{k}, \\ K̂_{0,−}(z) & = \frac{1}{2}, K̂_{ℓ,−}(z) = ∑_{k=0}^{ℓ} ĝ_{ℓ−k,−}z^{−k} = Ĝ_{ℓ,−}(z) + ĝ_{ℓ,−}, \\ K̂_{0,+}(z) & = 0, K̂_{ℓ,+}(z) = ∑_{k=1}^{ℓ} ĝ_{ℓ−k,+}z^{k} = Ĝ_{ℓ,+}(z) − ĝ_{ℓ,+}. \end{aligned} (2.38)

The stationary zero-curvature relation (2.29), 0 = UV\underline{p}−V + ; \underline{p}U, is then equivalent to

−α(g_{p_{+},+} + g_{p_{−},−}^{−}) + f_{p_{+}−1,+} − f_{p_{−}−1,−}^{−} = 0,  (2.39)
β(g_{p_{+},+}^{−} + g_{p_{−},−}) + h_{p_{+}−1,+}^{−} − h_{p_{−}−1,−} = 0.  (2.40)

1In this paper, a sum is interpreted as zero whenever the upper limit in the sum is strictly less than its lower limit.

Thus, varying p_{±} ∈ ℕ_{0}, equations (2.39) and (2.40) give rise to the stationary Ablowitz–Ladik (AL) hierarchy which we introduce as follows

s-AL\underline{p}(α, β) = (\begin{matrix}−α(g_{p_{+},+} + g_{p_{−},−}^{−}) + f_{p_{+}−1,+} − f_{p_{−}−1,−}^{−} \\ β(g_{p_{+},+}^{−} + g_{p_{−},−}) + h_{p_{+}−1,+}^{−} − h_{p_{−}−1,−} \end{matrix}) = 0, \underline{p} ∈ ℕ_{0}^{2}. (2.41)

Explicitly (recalling γ = 1 − αβ and taking p_{−} = p_{+} for simplicity),

s-AL_{(0,0)}(α, β) = (\begin{matrix}−c_{(0,0)}α \\ c_{(0,0)}β \end{matrix}) = 0,
s-AL_{(1,1)}(α, β) = (\begin{matrix}−γ(c_{0,−}α^{−} + c_{0,+}α^{+}) − c_{(1,1)}α \\ γ(c_{0,+}β^{−} + c_{0,−}β^{+}) + c_{(1,1)}β \end{matrix}) = 0,

s-AL_{(2,2)}(α, β) = (\begin{matrix}−γ(c_{0,+}α^{++}γ^{+} + c_{0,−}α^{−−}γ^{−} − α(c_{0,+}α^{+}β^{−} + c_{0,−}α^{−}β^{+}) \\ −β(c_{0,−}(α^{−})^{2} + c_{0,+}(α^{+})^{2})) \end{matrix} ; \begin{matrix}γ(c_{0,−}β^{++}γ^{+} + c_{0,+}β^{−−}γ^{−} − β(c_{0,+}α^{+}β^{−} + c_{0,−}α^{−}β^{+}) \\ −α(c_{0,+}(β^{−})^{2} + c_{0,−}(β^{+})^{2})) \end{matrix})

+ (\begin{matrix}−γ(c_{1,−}α^{−} + c_{1,+}α^{+}) − c_{(2,2)}α \\ γ(c_{1,+}β^{−} + c_{1,−}β^{+}) + c_{(2,2)}β \end{matrix}) = 0, etc.,  (2.42)

represent the first few equations of the stationary Ablowitz–Ladik hierarchy. Here we introduced

c\underline{p} = (c_{p_{−},−} + c_{p_{+},+})/2, p_{±} ∈ ℕ_{0}.  (2.43)

By definition, the set of solutions of (2.41), with \underline{p} ranging in ℕ_{0}^{2} and c_{ℓ,±} ∈ ℂ, ℓ ∈ ℕ_{0}, represents the class of algebro-geometric Ablowitz–Ladik solutions.

Using (2.1), one can show (cf. [25]) that g_{p_{+},+} = g_{p_{−},−} up to a lattice constant which can be set equal to zero without loss of generality. Thus, we will henceforth assume that

g_{p_{+},+} = g_{p_{−},−},  (2.44)

which in turn implies that

K\underline{p} = G\underline{p}  (2.45)

and hence renders V\underline{p} in (2.28) traceless in the stationary context. (We note that equations (2.44) and (2.45) cease to be valid in the time-dependent context, though.)

Next we turn to the time-dependent Ablowitz–Ladik hierarchy. For that purpose the coefficients α and β are now considered as functions of both the lattice point and time. For each equation in the hierarchy, that is, for each \underline{p}, we introduce a deformation (time) parameter t\underline{p} ∈ ℝ in α, β, replacing α(n), β(n) by α(n, t\underline{p}), β(n, t\underline{p}). Moreover, the definitions (2.27), (2.28), and (2.34)–(2.37) of U, V\underline{p}, and F\underline{p}, G\underline{p}, H\underline{p}, K\underline{p}, respectively, still apply. Imposing the zero-curvature relation

U_{t\underline{p}} + UV\underline{p} − V + ; \underline{p}U = 0, \underline{p} ∈ ℕ_{0}^{2},  (2.46)

then results in the equations

α_{t\underline{p}} = i(zF − ; \underline{p} + α(G\underline{p} + K− ; \underline{p}) − F\underline{p}),  (2.47)
β_{t\underline{p}} = −i(β(G− ; \underline{p} + K\underline{p}) − H\underline{p} + z^{−1}H− ; \underline{p}),  (2.48)
0 = z(G− ; \underline{p} − G\underline{p}) + zβF\underline{p} + αH− ; \underline{p},  (2.49)
0 = zβF − ; \underline{p} + αH\underline{p} + K− ; \underline{p} − K\underline{p}.  (2.50)

Varying \underline{p} ∈ ℕ_{0}^{2}, the collection of evolution equations

\begin{matrix}AL\underline{p}(α, β) = (\begin{matrix}−iα_{t\underline{p}} − α(g_{p_{+},+} + g_{p_{−},−}^{−}) + f_{p_{+}−1,+} − f_{p_{−}−1,−}^{−} \\ −iβ_{t\underline{p}} + β(g_{p_{+},+}^{−} + g_{p_{−},−}) − h_{p_{−}−1,−} + h_{p_{+}−1,+}^{−} \end{matrix}) = 0, \\ t\underline{p} ∈ ℝ, \underline{p} ∈ ℕ_{0}^{2}, \end{matrix} (2.51)

then defines the time-dependent Ablowitz–Ladik hierarchy. Explicitly, taking p_{−} = p_{+} for simplicity,

AL_{(0,0)}(α, β) = (\begin{matrix}−iα_{t_{(0,0)}} − c_{(0,0)}α \\ −iβ_{t_{(0,0)}} + c_{(0,0)}β \end{matrix}) = 0,

AL_{(1,1)}(α, β) = (\begin{matrix}−iα_{t_{(1,1)}} − γ(c_{0,−}α^{−} + c_{0,+}α^{+}) − c_{(1,1)}α \\ −iβ_{t_{(1,1)}} + γ(c_{0,+}β^{−} + c_{0,−}β^{+}) + c_{(1,1)}β \end{matrix}) = 0,

AL_{(2,2)}(α, β)

= (\begin{matrix}−iα_{t_{(2,2)}} − γ(c_{0,+}α^{++}γ^{+} + c_{0,−}α^{−−}γ^{−} − α(c_{0,+}α^{+}β^{−} + c_{0,−}α^{−}β^{+}) \\ −β(c_{0,−}(α^{−})^{2} + c_{0,+}(α^{+})^{2})) \end{matrix} ; \begin{matrix}−iβ_{t_{(2,2)}} + γ(c_{0,−}β^{++}γ^{+} + c_{0,+}β^{−−}γ^{−} − β(c_{0,+}α^{+}β^{−} + c_{0,−}α^{−}β^{+}) \\ −α(c_{0,+}(β^{−})^{2} + c_{0,−}(β^{+})^{2})) \end{matrix})

+ (\begin{matrix}−γ(c_{1,−}α^{−} + c_{1,+}α^{+}) − c_{(2,2)}α \\ γ(c_{1,+}β^{−} + c_{1,−}β^{+}) + c_{(2,2)}β \end{matrix}) = 0, etc.,  (2.52)

represent the first few equations of the time-dependent Ablowitz–Ladik hierarchy. Here we recall the definition of c\underline{p} in (2.43).

By (2.51), (2.6), and (2.10), the time derivative of γ = 1 − αβ is given by

γ_{t\underline{p}} = iγ((g_{p_{+},+} − g_{p_{+},+}^{−}) − (g_{p_{−},−} − g_{p_{−},−}^{−})),  (2.53)

or, alternatively,

γ_{t\underline{p}} = iγ(αz^{−1}H− ; \underline{p} − αH\underline{p} + βF\underline{p} − zβF − ; \underline{p}),  (2.54)

using (2.47)–(2.50).

Remark 2.5. (i) The special choices β = ±\overline{α}, c_{0,±} = 1 lead to the discrete nonlinear Schrödinger hierarchy. In particular, choosing c_{(1,1)} = −2 yields the discrete nonlinear Schrödinger equation in its usual form (see, e.g., [7, Ch. 3] and the references cited therein),

− iα_{t} − (1 ∓ |α|^{2})(α^{−} + α^{+}) + 2α = 0,  (2.55)

as its first nonlinear element. The choice β = \overline{α} is called the defocusing case, β = −\overline{α} represents the focusing case of the discrete nonlinear Schrödinger hierarchy. (ii) The alternative choice β = \overline{α}, c_{0,±} = ∓i, leads to the hierarchy of Schur flows. In particular, choosing c_{(1,1)} = 0 yields

α_{t} − (1 − |α|^{2})(α^{+} − α^{−}) = 0  (2.56)

as the first nonlinear element of this hierarchy (cf. [9], [16], [17], [29], [35], [44]).

3. Lax pairs for the AL hierarchy

In this section we introduce Lax pairs for the AL hierarchy and prove the equivalence of the zero-curvature and Lax representation. This result is new.

Throughout this section we suppose Hypothesis 2.1. We start by relating the homogeneous coefficients f̂_{ℓ,±}, ĝ_{ℓ,±}, and ĥ_{ℓ,±} to certain matrix elements of L, where L will later be identified as the Lax difference expression associated with the Ablowitz–Ladik hierarchy. For this purpose it is useful to introduce the standard basis {δ_{m}}_{m∈ℤ} in ℓ^{2}(ℤ) by

δ_{m} = \{δ_{m,n}\}_{n∈ℤ}, m ∈ ℤ, δ_{m,n} = \begin{cases}1, & m = n, \\ 0, & m ≠ n. \end{cases}  (3.1)

The scalar product in ℓ^{2}(ℤ), denoted by ( · , ·), is defined by

(f, g) = ∑_{n∈ℤ} \overline{f(n)}g(n), f, g ∈ ℓ^{2}(ℤ).  (3.2)

In the standard basis just defined, we introduce the difference expression L by

L = ( \begin{matrix}... & ... & ... & ... & ... & 0 \\ 0 & −α(0)ρ(−1) & −β(−1)α(0) & −α(1)ρ(0) & ρ(0)ρ(1) \\ ρ(−1)ρ(0) & β(−1)ρ(0) & −β(0)α(1) & β(0)ρ(1) & 0 \\ 0 & −α(2)ρ(1) & −β(1)α(2) & −α(3)ρ(2) & ρ(2)ρ(3) \\ 0 & ρ(1)ρ(2) & β(1)ρ(2) & −β(2)α(3) & β(2)ρ(3) & 0 \\ ... & ... & ... & ... & ... \end{matrix} ) (3.3)

= ( − β(n)α(n + 1)δ_{m,n}

+ (β(n − 1)ρ(n)δ_{odd}(n) − α(n + 1)ρ(n)δ_{even}(n))δ_{m,n−1}
+ (β(n)ρ(n + 1)δ_{odd}(n) − α(n + 2)ρ(n + 1)δ_{even}(n))δ_{m,n+1}  (3.4)
+ ρ(n + 1)ρ(n + 2)δ_{even}(n)δ_{m,n+2} + ρ(n − 1)ρ(n)δ_{odd}(n)δ_{m,n−2})_{m,n∈ℤ}

= ρ^{−}ρ δ_{even} S^{−−} + (β^{−}ρ δ_{even} − α^{+}ρ δ_{odd})S^{−} − βα^{+}

+ (βρ^{+} δ_{even} − α^{++}ρ^{+} δ_{odd})S^{+} + ρ^{+}ρ^{++} δ_{odd} S^{++},  (3.5)

where δ_{even} and δ_{odd} denote the characteristic functions of the even and odd integers,

δ_{even} = χ_{2ℤ}, δ_{odd} = 1 − δ_{even} = χ_{2ℤ+1}.  (3.6)

In particular, terms of the form −β(n)α(n+1) represent the diagonal (n, n)-entries, n ∈ ℤ, in the infinite matrix (3.3). In addition, we used the abbreviation

ρ = γ^{1/2} = (1 − αβ)^{1/2}.  (3.7)

Next, we introduce the unitary operator Uε̃ in ℓ^{2}(ℤ) by

Uε̃ = (ε̃(n)δ_{m,n})_{(m,n)∈ℤ^{2}}, ε̃(n) ∈ {1, −1}, n ∈ ℤ,  (3.8)

and the sequence ε = {ε(n)}_{n∈ℤ} ∈ ℂ^{ℤ} by

ε(n) = ε̃(n − 1)ε̃(n), n ∈ ℤ.  (3.9)

Assuming α, β ∈ ℓ^{∞}(ℤ), a straightforward computation then shows that

L̆_{ε} = Uε̃L̆U −1 ; ε̃,  (3.10)

where L_{ε} is associated with the sequences α_{ε} = α, β_{ε} = β, and ρ_{ε} = ερ, and L̆ and L̆_{ε} are the bounded operator realizations of L and L_{ε} in ℓ^{2}(ℤ), respectively. Moreover, the recursion formalism in (2.5)–(2.12) yields coefficients which are polynomials in α, β and some of their shifts and hence depends only quadratically on ρ. As a result, the choice of square root of ρ(n), n ∈ ℤ, in (3.7) is immaterial when introducing the AL hierarchy via the Lax equations (3.38).

The half-lattice (i.e., semi-infinite) version of L was recently rediscovered by Cantero, Moral, and Velázquez [11] in the special case where β = \overline{α} (see also Simon [42], [43] who coined the term CMV matrix in this context). The matrix representation of L^{−1} is then obtained from that of L in (3.3) by taking the formal adjoint of L and subsequently exchanging α and β

L^{−1} = ( − α(n)β(n + 1)δ_{m,n} + (α(n − 1)ρ(n)δ_{even}(n)  (3.11)
− β(n + 1)ρ(n)δ_{odd}(n))δ_{m,n−1}
+ (α(n)ρ(n + 1)δ_{even}(n) − β(n + 2)ρ(n + 1)δ_{odd}(n))δ_{m,n+1}

+ ρ(n + 1)ρ(n + 2)δ_{odd}(n)δ_{m,n+2} + ρ(n − 1)ρ(n)δ_{even}(n)δ_{m,n−2})_{m,n∈ℤ}

= ρ^{−}ρ δ_{odd} S^{−−} + (α^{−}ρ δ_{odd} − β^{+}ρ δ_{even})S^{−} − αβ^{+}
+ (αρ^{+} δ_{odd} − β^{++}ρ^{+} δ_{even})S^{+} + ρ^{+}ρ^{++} δ_{even} S^{++}.  (3.12)

L and L^{−1} define bounded operators in ℓ^{2}(ℤ) if α and β are bounded sequences. However, this is of no importance in the context of Lemma 3.1 below as we only apply the five-diagonal matrices L and L^{−1} to basis vectors of the type δ_{m}.

Next, we discuss a useful factorization of L. For this purpose we introduce the sequence of 2 × 2 matrices θ(n), n ∈ ℤ, by

θ(n) = (\begin{matrix}−α(n) & ρ(n) \\ ρ(n) & β(n) \end{matrix}) , n ∈ ℤ,  (3.13)

and two difference expressions D and E by their matrix representations in the standard basis (3.1) of ℓ^{2}(ℤ)

D = ( \begin{matrix}... & 0 \\ θ(2n−2) \\ & θ(2n) \\ 0 & ... \end{matrix} ) , E = ( \begin{matrix}... & 0 \\ θ(2n−1) \\ & θ(2n+1) \\ 0 & ... \end{matrix} ) ,  (3.14)

where

\begin{aligned}(\begin{matrix}D(2n − 1, 2n − 1) & D(2n − 1, 2n) \\ D(2n, 2n − 1) & D(2n, 2n) \end{matrix}) & = θ(2n), \\ (\begin{matrix}E(2n, 2n) & E(2n, 2n + 1) \\ E(2n + 1, 2n) & E(2n + 1, 2n + 1) \end{matrix}) & = θ(2n + 1), n ∈ ℤ. \end{aligned}  (3.15)

Then L can be factorized into

L = DE.  (3.16)

Explicitly, D and E are given by

D = ρ δ_{even} S^{−} − α^{+} δ_{odd} + β δ_{even} + ρ^{+} δ_{odd} S^{+},  (3.17)

E = ρ δ_{odd} S^{−} + β δ_{odd} − α^{+} δ_{even} + ρ^{+} δ_{even} S^{+},  (3.18)

and their inverses are of the form

D^{−1} = ρ δ_{even} S^{−} − β^{+} δ_{odd} + α δ_{even} + ρ^{+} δ_{odd} S^{+},  (3.19)
E^{−1} = ρ δ_{odd} S^{−} + α δ_{odd} − β^{+} δ_{even} + ρ^{+} δ_{even} S^{+}.  (3.20)

The next result details the connections between L and the recursion coefficients f_{ℓ,±}, g_{ℓ,±}, and h_{ℓ,±}.

Lemma 3.1. Let n ∈ ℤ. Then the homogeneous coefficients {f̂_{ℓ,±}}_{ℓ∈ℕ_{0}}, {ĝ_{ℓ,±}}_{ℓ∈ℕ_{0}}, and {ĥ_{ℓ,±}}_{ℓ∈ℕ_{0}} satisfy the following relations:

f̂_{ℓ,+}(n) = (δ_{n}, EL^{ℓ}δ_{n})δ_{even}(n) + (δ_{n}, L^{ℓ}Dδ_{n})δ_{odd}(n), ℓ ∈ ℕ_{0},
f̂_{ℓ,−}(n) = (δ_{n}, D^{−1}L^{−ℓ}δ_{n})δ_{even}(n) + (δ_{n}, L^{−ℓ}E^{−1}δ_{n})δ_{odd}(n), ℓ ∈ ℕ_{0},
ĝ_{0,±} = 1/2, ĝ_{ℓ,±}(n) = (δ_{n}, L^{±ℓ}δ_{n}), ℓ ∈ ℕ,  (3.21)
ĥ_{ℓ,+}(n) = (δ_{n}, L^{ℓ}Dδ_{n})δ_{even}(n) + (δ_{n}, EL^{ℓ}δ_{n})δ_{odd}(n), ℓ ∈ ℕ_{0},
ĥ_{ℓ,−}(n) = (δ_{n}, L^{−ℓ}E^{−1}δ_{n})δ_{even}(n) + (δ_{n}, D^{−1}L^{−ℓ}δ_{n})δ_{odd}(n) ℓ ∈ ℕ_{0}.

Proof. Using (3.16)–(3.20) we show that the sequences defined in (3.21) satisfy the recursion relations of Lemma 2.4 respectively relation (2.6). For n even,

\begin{aligned}ĝ_{ℓ,+}(n) − ĝ_{ℓ,+}(n − 1) & = (δ_{n}, DEL^{ℓ−1}δ_{n}) − (δ_{n−1}, DEL^{ℓ−1}δ_{n−1}) \\ & = (D^{∗}δ_{n}, EL^{ℓ−1}δ_{n}) − (D^{∗}δ_{n−1}, EL^{ℓ−1}δ_{n−1}) \\ & = β(n)(δ_{n}, EL^{ℓ−1}δ_{n}) + ρ(n)(δ_{n−1}, EL^{ℓ−1}δ_{n}) \\ & + α(n)(δ_{n−1}, EL^{ℓ−1}δ_{n−1}) − ρ(n)(δ_{n}, EL^{ℓ−1}δ_{n−1}) \\ & = β(n)f̂_{ℓ−1,+}(n) + α(n)ĥ_{ℓ−1,+}(n − 1), \end{aligned} (3.22)

since (EL^{ℓ})^{⊤} = EL^{ℓ} by (3.13), (3.16). Moreover,

\begin{aligned}f̂_{ℓ,+}(n) & = (δ_{n}, EL^{ℓ}δ_{n}) = (E^{∗}δ_{n}, L^{ℓ}δ_{n}) \\ & = −α(n + 1)(δ_{n}, L^{ℓ}δ_{n}) + ρ(n + 1)(δ_{n+1}, L^{ℓ}δ_{n}) \\ & + α(n + 1)(δ_{n+1}, L^{ℓ}δ_{n+1}) − α(n + 1)(δ_{n+1}, L^{ℓ}δ_{n+1}) \\ & = f̂_{ℓ−1,+}(n + 1) − α(n + 1)(ĝ_{ℓ,+}(n + 1) + ĝ_{ℓ,+}(n)), \\ ĥ_{ℓ,+}(n) & = (δ_{n}, L^{ℓ}Dδ_{n}) = β(n)(δ_{n}, L^{ℓ}δ_{n}) + ρ(n)(δ_{n}, L^{ℓ}δ_{n−1}) \\ & + β(n)(δ_{n−1}, L^{ℓ}δ_{n−1}) − β(n)(δ_{n−1}, L^{ℓ}δ_{n−1}) \\ & = ĥ_{ℓ−1,+}(n − 1) + β(n)(ĝ_{ℓ,+}(n) + ĝ_{ℓ,+}(n − 1)), \end{aligned}  (3.23)

that is, the coefficients satisfy (2.23). The remaining cases follow analogously. □

Finally, we derive an explicit expression for the Lax pair for the Ablowitz–Ladik hierarchy, but first we need some notation. Let T be a bounded operator in ℓ^{2}(ℤ). Given the standard basis (3.1) in ℓ^{2}(ℤ), we represent T by

T = (T (m, n))_{(m,n)∈ℤ^{2}}, T (m, n) = (δ_{m}, T δ_{n}), (m, n) ∈ ℤ^{2}.  (3.24)

Actually, for our purpose below, it is sufficient that T is an N-diagonal matrix for some N ∈ ℕ. Moreover, we introduce the upper and lower triangular parts T_{±} of T by

T_{±} = (T_{±}(m, n))_{(m,n)∈ℤ^{2}}, T_{±}(m, n) = \begin{cases}T (m, n), & ±(n − m) > 0, \\ 0, & otherwise. \end{cases} (3.25)

Next, consider the finite difference expression P\underline{p} defined by

P\underline{p} = \frac{i}{2} ∑_{ℓ=1}^{p_{+}} c_{p_{+}−ℓ,+}((L^{ℓ})_{+} − (L^{ℓ})_{−}) − \frac{i}{2} ∑_{ℓ=1}^{p_{−}} c_{p_{−}−ℓ,−}((L^{−ℓ})_{+} − (L^{−ℓ})_{−}) − \frac{i}{2}c\underline{p} Q_{d},
\underline{p} ∈ ℕ_{0}^{2},  (3.26)

with L given by (3.3) and Q_{d} denoting the doubly infinite diagonal matrix

Q_{d} = ((−1)^{k}δ_{k,ℓ})_{k,ℓ∈ℤ}.  (3.27)

Before we prove that (L, P\underline{p}) is indeed the Lax pair for the Ablowitz–Ladik hierarchy, we derive one more representation of P\underline{p} in terms of L.

We denote by ℓ_{0}(ℤ) the set of complex-valued sequences of compact support. If R denotes a finite difference expression, then ψ is called a weak solution of Rψ = zψ, for some z ∈ ℂ, if the relation holds pointwise for each lattice point, that is, if ((R − z)ψ)(n) = 0 for all n ∈ ℤ.

Lemma 3.2. Let ψ ∈ ℓ_{0}^{∞}(ℤ). Then the difference expression P\underline{p} defined in (3.26) acts on ψ by

(P\underline{p}ψ)(n) = i( − ∑_{ℓ=1}^{p_{−}} f_{p_{−}−ℓ,−}(n)(EL^{−ℓ}ψ)(n) − ∑_{ℓ=0}^{p_{+}−1} f_{p_{+}−1−ℓ,+}(n)(EL^{ℓ}ψ)(n)

+ ∑_{ℓ=1}^{p_{−}} g_{p_{−}−ℓ,−}(n)(L^{−ℓ}ψ)(n) + ∑_{ℓ=1}^{p_{+}} g_{p_{+}−ℓ,+}(n)(L^{ℓ}ψ)(n)

+ \frac{1}{2}(g_{p_{−},−}(n) + g_{p_{+},+}(n))ψ(n))δ_{odd}(n)  (3.28)

+ i( ∑_{ℓ=0}^{p_{−}−1} h_{p_{−}−1−ℓ,−}(n)(D^{−1}L^{−ℓ}ψ)(n) + ∑_{ℓ=1}^{p_{+}} h_{p_{+}−ℓ,+}(n)(D^{−1}L^{ℓ}ψ)(n)

− ∑_{ℓ=1}^{p_{−}} g_{p_{−}−ℓ,−}(n)(L^{−ℓ}ψ)(n) − ∑_{ℓ=1}^{p_{+}} g_{p_{+}−ℓ,+}(n)(L^{ℓ}ψ)(n)

− \frac{1}{2}(g_{p_{−},−}(n) + g_{p_{+},+}(n))ψ(n))δ_{even}(n), n ∈ ℤ.

In addition, if u is a weak solution of Lu(z) = zu(z), then

(P\underline{p}u(z))(n)

= ( − iF\underline{p}(z, n)(Eu(z))(n) + \frac{i}{2}(G\underline{p}(z, n) + K\underline{p}(z, n))u(z, n))δ_{odd}(n)

+ (iH\underline{p}(z, n)(D^{−1}u(z))(n) − \frac{i}{2}(G\underline{p}(z, n) + K\underline{p}(z, n))u(z, n))δ_{even}(n),

n ∈ ℤ, (3.29)

in the weak sense.

14 F. GESZTESY, H. HOLDEN, J. MICHOR, AND G. TESCHL

Proof. We consider the case where n is even and use induction on \underline{p} = (p_{−}, p_{+}). The case n odd is analogous. For \underline{p} = (0, 0), the formulas (3.28) and (3.26) match. Denoting by P̂\underline{p} the corresponding homogeneous operator where all summation constants c_{k,±}, k = 1, …, p_{±}, vanish, we have to show that

\begin{aligned}iP̂\underline{p} & = iP̂_{p_{+}−1}^{+}L − ĥ_{p_{+}−1,+}D^{−1}L + \frac{1}{2}(ĝ_{p_{+}−1,+}L + ĝ_{p_{+},+}) \\ & + iP̂_{p_{−}−1}^{−}L^{−1} − ĥ_{p_{−}−1,−}D^{−1} + \frac{1}{2}(ĝ_{p_{−}−1,−}L^{−1} + ĝ_{p_{−},−}), \end{aligned}  (3.30)

where P̂_{j}^{±} correspond to the powers of L in (3.26), P̂_{j}^{±} = \frac{i}{2}((L^{±j})_{±}−(L^{±j})_{∓}). This can be done upon considering (δ_{m}, P̂\underline{p}δ_{n}) and making appropriate case distinctions m = n, m > n, and m < n.

Using (3.4), (3.11), (3.16)–(3.20), (3.25), and Lemma 3.1, one verifies for instance in the case m = n, (δ_{n}, iP̂_{p_{+}}^{+}δ_{n}) = (δ_{n}, iP̂_{p_{+}−1}^{+}Lδ_{n}) + α(n + 1)ĥ_{p_{+}−1,+}(n)

+ \frac{1}{2}(ĝ_{p_{+},+}(n) − α(n + 1)β(n)ĝ_{p_{+}−1,+}(n))

= (δ_{n}, \frac{1}{2}((L^{p_{+}−1})_{+} − (L^{p_{+}−1})_{−})(α^{++}ρ^{+}δ_{n−1} + α^{+}βδ_{n} + α^{+}ρδ_{n+1} − ρ^{−}ρδ_{n+2}))

+ α(n + 1)ĥ_{p_{+}−1,+}(n) + \frac{1}{2}(ĝ_{p_{+},+}(n) − α(n + 1)β(n)ĝ_{p_{+}−1,+}(n))

= −\frac{1}{2}α(n + 1)ρ(n)(δ_{n}, L^{p_{+}−1}δ_{n−1}) + \frac{1}{2}α(n + 2)ρ(n + 1)(δ_{n}, L^{p_{+}−1}δ_{n+1})

− \frac{1}{2}ρ(n + 1)ρ(n + 2)(δ_{n}, L^{p_{+}−1}δ_{n+2}) + α(n + 1)ĥ_{p_{+}−1,+}(n)

+ \frac{1}{2}(ĝ_{p_{+},+}(n) − α(n + 1)β(n)ĝ_{p_{+}−1,+}(n))

= −α(n + 1)ρ(n)(δ_{n}, L^{p_{+}−1}δ_{n−1})

− α(n + 1)β(n)ĝ_{p_{+}−1,+}(n) + α(n + 1)ĥ_{p_{+}−1,+}(n)

= 0,  (3.31)

since by Lemma 3.1,

\begin{aligned}ĝ_{p_{+},+}(n) & = (δ_{n}, L^{p_{+}−1}Lδ_{n}), \\ ĥ_{p_{+}−1,+}(n) & = (δ_{n}, L^{p_{+}−1}Dδ_{n}) = β(n)(δ_{n}, L^{p_{+}−1}δ_{n}) + ρ(n)(δ_{n}, L^{p_{+}−1}δ_{n−1}). \end{aligned}

Similarly, (δ_{n}, iP̂_{p_{−}}^{−}δ_{n})

= (δ_{n}, iP̂_{p_{−}−1}^{−}L^{−1}δ_{n}) − α(n)ĥ_{p_{−}−1,−}(n) + \frac{1}{2}(ĝ_{p_{−},−}(n) − α(n)β(n + 1)ĝ_{p_{−}−1,−}(n))

= (δ_{n}, \frac{1}{2}((L^{1−p_{−}})_{+} − (L^{1−p_{−}})_{−})(ρ^{+}ρ^{++}δ_{n−2} + αρ^{+}δ_{n−1} − αβ^{+}δ_{n} + α^{−}ρδ_{n+1}))

− α(n)ĥ_{p_{−}−1,−}(n) + \frac{1}{2}(ĝ_{p_{−},−}(n) − α(n)β(n + 1)ĝ_{p_{−}−1,−}(n))

= −\frac{1}{2}ρ(n − 1)ρ(n)(δ_{n}, L^{1−p_{−}}δ_{n−2}) − \frac{1}{2}α(n − 1)ρ(n)(δ_{n}, L^{1−p_{−}}δ_{n−1})

+ \frac{1}{2}α(n)ρ(n + 1)(δ_{n}, L^{1−p_{−}}δ_{n+1}) − α(n)ĥ_{p_{−}−1,−}(n)

+ \frac{1}{2}(ĝ_{p_{−},−}(n) − α(n)β(n + 1)ĝ_{p_{−}−1,−}(n))

= α(n)ρ(n + 1)(δ_{n}, L^{1−p_{−}}δ_{n+1}) − α(n)β(n + 1)ĝ_{p_{−}−1,−}(n) − α(n)ĥ_{p_{−}−1,−}(n)

= 0,  (3.32)

where we used Lemma 3.1 and (2.12) at ℓ = p_{−} − 2 for the last equality. This proves the case m = n. The remaining cases m > n and m < n are settled in a similar fashion.

Equality (3.29) then follows from Lu(z) = zu(z) and (2.34)–(2.37). □

Next, we introduce the difference expression P ⊤ ; \underline{p} by

\begin{aligned}P ⊤ ; \underline{p} & = −\frac{i}{2} ∑_{ℓ=1}^{p_{+}} c_{p_{+}−ℓ,+}(((L^{⊤})^{ℓ})_{+} − ((L^{⊤})^{ℓ})_{−}) \\ & + \frac{i}{2} ∑_{ℓ=1}^{p_{−}} c_{p_{−}−ℓ,−}(((L^{⊤})^{−ℓ})_{+} − ((L^{⊤})^{−ℓ})_{−}) − \frac{i}{2}c\underline{p} Q_{d}, \underline{p} ∈ ℕ_{0}^{2}, \end{aligned} (3.33)

with L^{⊤} = ED the difference expression associated with the transpose of the infinite matrix (3.3) in the standard basis of ℓ^{2}(ℤ) and Q_{d} denoting the doubly infinite diagonal matrix in (3.27). Here we used

(M_{+})^{⊤} = (M^{⊤})_{−}, (M_{−})^{⊤} = (M^{⊤})_{+}  (3.34)

for a finite difference expression M in the standard basis of ℓ^{2}(ℤ).

For later purpose in Section 5 we now mention the analog of Lemma 3.2 for the difference expression P ⊤ ; \underline{p} without proof:

Lemma 3.3. Let χ ∈ ℓ_{0}^{∞}(ℤ). Then the difference expression P ⊤ ; \underline{p} defined in (3.33) acts on χ by

(P ⊤ ; \underline{p}χ)(n) = i( − ∑_{ℓ=0}^{p_{−}−1} h_{p_{−}−1−ℓ,−}(n)(E^{−1}(L^{⊤})^{−ℓ}χ)(n)

− ∑_{ℓ=1}^{p_{+}} h_{p_{+}−ℓ,+}(n)(E^{−1}(L^{⊤})^{ℓ}χ)(n)

+ ∑_{ℓ=1}^{p_{−}} g_{p_{−}−ℓ,−}(n)((L^{⊤})^{−ℓ}χ)(n) + ∑_{ℓ=1}^{p_{+}} g_{p_{+}−ℓ,+}(n)((L^{⊤})^{ℓ}χ)(n)

+ \frac{1}{2}(g_{p_{−},−}(n) + g_{p_{+},+}(n))χ(n))δ_{odd}(n)  (3.35)

+ i( ∑_{ℓ=1}^{p_{−}} f_{p_{−}−ℓ,−}(n)(D(L^{⊤})^{−ℓ}χ)(n)

+ ∑_{ℓ=0}^{p_{+}−1} f_{p_{+}−1−ℓ,+}(n)(D(L^{⊤})^{ℓ}χ)(n)

− ∑_{ℓ=1}^{p_{−}} g_{p_{−}−ℓ,−}(n)((L^{⊤})^{−ℓ}χ)(n) − ∑_{ℓ=1}^{p_{+}} g_{p_{+}−ℓ,+}(n)((L^{⊤})^{ℓ}χ)(n)

− \frac{1}{2}(g_{p_{−},−}(n) + g_{p_{+},+}(n))χ(n))δ_{even}(n), n ∈ ℤ.

In addition, if χ is a weak solution of L^{⊤}v(z) = zv(z), then

(P ⊤ ; \underline{p}v(z))(n)

= −i(H\underline{p}(z, n)(E^{−1}v(z))(n) − \frac{1}{2}(G\underline{p}(z, n) + K\underline{p}(z, n))v(z, n))δ_{odd}(n)

+ i(F\underline{p}(z, n)(Dv(z))(n) − \frac{1}{2}(G\underline{p}(z, n) + K\underline{p}(z, n))v(z, n))δ_{even}(n),

n ∈ ℤ, (3.36)

in the weak sense.

Given these preliminaries, one can now prove the following result, the proof of which is based on fairly tedious computations. We present them here in some detail as these results have not appeared in print before.

Theorem 3.4. Assume Hypothesis 2.1. Then, for each \underline{p} ∈ ℕ_{0}^{2}, the \underline{p}th stationary Ablowitz–Ladik equation s-AL\underline{p}(α, β) = 0 in (2.41) is equivalent to the vanishing of the commutator of P\underline{p} and L,

[P\underline{p}, L] = 0.  (3.37)

In addition, the \underline{p}th time-dependent Ablowitz–Ladik equation AL\underline{p}(α, β) = 0 in (2.51) is equivalent to the Lax commutator equations

L_{t\underline{p}}(t\underline{p}) − [P\underline{p}(t\underline{p}), L(t\underline{p})] = 0, t\underline{p} ∈ ℝ.  (3.38)

In particular, the pair of difference expressions (L, P\underline{p}) represents the Lax pair for the Ablowitz–Ladik hierarchy of nonlinear differential-difference evolution equations.

Proof. Let f ∈ ℓ_{0}(ℤ). To curb the length of this proof we will only consider the case n even. We apply formulas (3.28) to compute the commutator ([P\underline{p}, L]f)(n) by rewriting D^{−1}L^{ℓ} = EL^{ℓ−1} and using (3.5), (3.17), and (3.18). This yields

i([P\underline{p}, L]f)(n)

= ( ∑_{ℓ=1}^{p_{+}} ρ^{−}ρ(g_{p_{+}−ℓ,+}^{−} − g_{p_{+}−ℓ,+}^{−−})

− ∑_{ℓ=0}^{p_{+}−1} ρ^{−}ρ(α^{−}h_{p_{+}−1−ℓ,+}^{−−} + β^{−}f_{p_{+}−1−ℓ,+}^{−}))(L^{ℓ}f)(n − 2)

+ ( ∑_{ℓ=1}^{p_{+}} ρ(2β^{−}g_{p_{+}−ℓ,+}^{−} − h_{p_{+}−ℓ,+}^{−})

+ ∑_{ℓ=0}^{p_{+}−1} ρ((ρ^{−})^{2}h_{p_{+}−1−ℓ,+}^{−−} − (β^{−})^{2}f_{p_{+}−1−ℓ,+}^{−}))(L^{ℓ}f)(n − 1)

+ ( ∑_{ℓ=1}^{p_{+}} ρ^{+}(β(g_{p_{+}−ℓ,+}^{+} + g_{p_{+}−ℓ,+}^{−}) − h_{p_{+}−ℓ,+})

CONSERVATION LAWS AND HAMILTONIANS FOR THE AL HIERARCHY 17

+ ∑_{ℓ=0}^{p_{+}−1} ρ^{+}(h_{p_{+}−1−ℓ,+}^{−} − ββ^{+}f_{p_{+}−1−ℓ,+}^{+} − βα^{+}h_{p_{+}−1−ℓ,+}))(L^{ℓ}f)(n + 1)

+ ( ∑_{ℓ=2}^{p_{+}+1} (g_{p_{+}+1−ℓ,+} − g_{p_{+}+1−ℓ,+}^{−})

+ ∑_{ℓ=1}^{p_{+}} (α^{+}(β(g_{p_{+}−ℓ,+} − g_{p_{+}−ℓ,+}^{−}) + h_{p_{+}−ℓ,+}) − αh_{p_{+}−ℓ,+}^{−})

+ ∑_{ℓ=0}^{p_{+}−1} (β(α^{+})^{2}h_{p_{+}−1−ℓ,+} − β(ρ^{+})^{2}f_{p_{+}−1−ℓ,+}^{+} − α^{+}h_{p_{+}−1−ℓ,+}^{−}))(L^{ℓ}f)(n)

+ ∑_{ℓ=1}^{p_{−}} ρ^{−}ρ(g_{p_{−}−ℓ,−}^{−} − g_{p_{−}−ℓ,−}^{−−} − β^{−}f_{p_{−}−ℓ,−}^{−} − α^{−}h_{p_{−}−ℓ,−}^{−−})(L^{−ℓ}f)(n − 2)

+ ( ∑_{ℓ=1}^{p_{−}} ρ(β^{−}(2g_{p_{−}−ℓ,−}^{−} − β^{−}f_{p_{−}−ℓ,−}^{−}) + ρ(ρ^{−})^{2}h_{p_{−}−ℓ,−}^{−−})

− ∑_{ℓ=0}^{p_{−}−1} ρh_{p_{−}−1−ℓ,−}^{−})(L^{−ℓ}f)(n − 1)

+ ( ∑_{ℓ=1}^{p_{−}} (βρ^{+}(g_{p_{−}−ℓ,−}^{+} + g_{p_{−}−ℓ,−}^{−} − α^{+}h_{p_{−}−ℓ,−} − β^{+}f_{p_{−}−ℓ,−}^{+}) + ρ^{+}h_{p_{−}−ℓ,−}^{−})

− ∑_{ℓ=0}^{p_{−}−1} ρ^{+}h_{p_{−}−1−ℓ,−})(L^{−ℓ}f)(n + 1)

+ ( ∑_{ℓ=1}^{p_{−}} (βα^{+}(g_{p_{−}−ℓ,−} − g_{p_{−}−ℓ,−}^{−} + α^{+}h_{p_{−}−ℓ,−} + β^{+}f_{p_{−}−ℓ,−}^{+}) − βf_{p_{−}−ℓ,−}^{+}

− α^{+}h_{p_{−}−ℓ,−}^{−})

+ ∑_{ℓ=0}^{p_{−}−1} (g_{p_{−}−1−ℓ,−} − g_{p_{−}−1−ℓ,−}^{−} − αh_{p_{−}−1−ℓ,−}^{−} + α^{+}h_{p_{−}−1−ℓ,−}))(L^{−ℓ}f)(n)

+ \frac{1}{2}(βρ^{+}(g_{p_{−},−}^{+} + g_{p_{+},+}^{+} + g_{p_{−},−} + g_{p_{+},+})f(n + 1)

+ β^{−}ρ(g_{p_{−},−}^{−} + g_{p_{+},+}^{−} + g_{p_{−},−} + g_{p_{+},+})f(n − 1)

− ρ^{−}ρ(g_{p_{−},−}^{−−} + g_{p_{+},+}^{−−} − g_{p_{−},−} − g_{p_{+},+})f(n − 2)),  (3.39)

where we added the terms

0 = − ∑_{ℓ=1}^{p_{+}} g_{p_{+}−ℓ,+}^{−}(L^{ℓ+1}f)(n) + ∑_{ℓ=1}^{p_{+}} g_{p_{+}−ℓ,+}^{−}(L^{ℓ+1}f)(n)

= − ∑_{ℓ=2}^{p_{+}+1} g_{p_{+}+1−ℓ,+}^{−}(L^{ℓ}f)(n) + ∑_{ℓ=1}^{p_{+}} g_{p_{+}−ℓ,+}^{−}L(L^{ℓ}f)(n),

0 = − ∑_{ℓ=1}^{p_{+}} h_{p_{+}−ℓ,+}^{−}(D^{−1}L^{ℓ}f)(n) + ∑_{ℓ=1}^{p_{+}} h_{p_{+}−ℓ,+}^{−}(EL^{ℓ−1}f)(n)

= − ∑_{ℓ=1}^{p_{+}} h_{p_{+}−ℓ,+}^{−}(α(L^{ℓ}f)(n) + ρ(L^{ℓ}f)(n − 1))

+ ∑_{ℓ=0}^{p_{+}−1} h_{p_{+}−1−ℓ,+}^{−}( − α^{+}(L^{ℓ}f)(n) + ρ^{+}(L^{ℓ}f)(n + 1)),  (3.40)
0 = − ∑_{ℓ=1}^{p_{−}} g_{p_{−}−ℓ,−}^{−}(L^{−ℓ+1}f)(n) + ∑_{ℓ=1}^{p_{−}} g_{p_{−}−ℓ,−}^{−}(L^{−ℓ+1}f)(n)

= − ∑_{ℓ=0}^{p_{−}−1} g_{p_{−}−1−ℓ,−}^{−}(L^{−ℓ}f)(n) + ∑_{ℓ=1}^{p_{−}} g_{p_{−}−ℓ,−}^{−}L(L^{−ℓ}f)(n),

0 = − ∑_{ℓ=1}^{p_{−}} h_{p_{−}−ℓ,−}^{−}(D^{−1}L^{−ℓ+1}f)(n) + ∑_{ℓ=1}^{p_{−}} h_{p_{−}−ℓ,−}^{−}(EL^{−ℓ}f)(n)

= − ∑_{ℓ=0}^{p_{−}−1} h_{p_{−}−1−ℓ,−}^{−}(α(L^{−ℓ}f)(n) + ρ(L^{−ℓ}f)(n − 1))

+ ∑_{ℓ=1}^{p_{−}} h_{p_{−}−ℓ,−}^{−}( − α^{+}(L^{−ℓ}f)(n) + ρ^{+}(L^{−ℓ}f)(n + 1)).

Next we apply the recursion relations (2.5)–(2.12). In addition, we also use

α^{+}h_{p_{−}−ℓ,−} + β^{+}f_{p_{−}−ℓ,−}^{+} = α^{+}(h_{p_{−}−1−ℓ,−}^{+} − β^{+}(g_{p_{−}−ℓ,−}^{+} + g_{p_{−}−ℓ,−}))
+ β^{+}(f_{p_{−}−1−ℓ,−} + α^{+}(g_{p_{−}−ℓ,−}^{+} + g_{p_{−}−ℓ,−}))
= g_{p_{−}−ℓ,−}^{+} − g_{p_{−}−ℓ,−}.  (3.41)

This implies, i([P\underline{p}, L]f)(n)

= ∑_{ℓ=1}^{p_{+}−1} ρ^{−}ρ(g_{p_{+}−ℓ,+}^{−} − g_{p_{+}−ℓ,+}^{−−} − α^{−}h_{p_{+}−1−ℓ,+}^{−−} − β^{−}f_{p_{+}−1−ℓ,+}^{−})(L^{ℓ}f)(n − 2)

+ ∑_{ℓ=1}^{p_{+}−1} (β^{−}ρ(g_{p_{+}−ℓ,+}^{−} − g_{p_{+}−ℓ,+}^{−−} − α^{−}h_{p_{+}−1−ℓ,+}^{−−} − β^{−}f_{p_{+}−1−ℓ,+}^{−})

+ ρ(β^{−}(g_{p_{+}−ℓ,+}^{−} + g_{p_{+}−ℓ,+}^{−−}) + h_{p_{+}−1−ℓ,+}^{−−} − h_{p_{+}−ℓ,+}^{−}))(L^{ℓ}f)(n − 1)

+ ∑_{ℓ=1}^{p_{+}−1} (βρ^{+}(g_{p_{+}−ℓ,+}^{+} − g_{p_{+}−ℓ,+} − α^{+}h_{p_{+}−1−ℓ,+} − β^{+}f_{p_{+}−1−ℓ,+}^{+})

+ ρ^{+}(β(g_{p_{+}−ℓ,+} + g_{p_{+}−ℓ,+}^{−}) + h_{p_{+}−1−ℓ,+}^{−} − h_{p_{+}−ℓ,+}))(L^{ℓ}f)(n + 1)

+ ( ∑_{ℓ=1}^{p_{+}−1} (g_{p_{+}+1−ℓ,+} − g_{p_{+}+1−ℓ,+}^{−} − αh_{p_{+}−ℓ,+}^{−} + βα^{+}(g_{p_{+}−ℓ,+}^{+} + g_{p_{+}−ℓ,+})

CONSERVATION LAWS AND HAMILTONIANS FOR THE AL HIERARCHY 19

− βf_{p_{+}−1−ℓ,+}^{+})  (3.42)

+ ∑_{ℓ=1}^{p_{+}−1} α^{+}(β(−g_{p_{+}−ℓ,+} − g_{p_{+}−ℓ,+}^{−}) + h_{p_{+}−ℓ,+} − h_{p_{+}−1−ℓ,+}^{−})

+ ∑_{ℓ=0}^{p_{+}−1} βα^{+}(g_{p_{+}−ℓ,+} − g_{p_{+}−ℓ,+}^{+} + α^{+}h_{p_{+}−1−ℓ,+} + β^{+}f_{p_{+}−1−ℓ,+}^{+}))(L^{ℓ}f)(n)

+ ∑_{ℓ=1}^{p_{−}} ρ^{−}ρ(g_{p_{−}−ℓ,−}^{−} − g_{p_{−}−ℓ,−}^{−−} − β^{−}f_{p_{−}−ℓ,−}^{−} − α^{−}h_{p_{−}−ℓ,−}^{−−})(L^{−ℓ}f)(n − 2)

+ ∑_{ℓ=1}^{p_{−}−1} (β^{−}ρ(g_{p_{−}−ℓ,−}^{−} − g_{p_{−}−ℓ,−}^{−−} − β^{−}f_{p_{−}−ℓ,−}^{−} − α^{−}h_{p_{−}−ℓ,−}^{−−})

+ ρ(β^{−}(g_{p_{−}−ℓ,−}^{−} + g_{p_{−}−ℓ,−}^{−−}) + h_{p_{−}−ℓ,−}^{−−} − h_{p_{−}−1−ℓ,−}^{−}))(L^{−ℓ}f)(n − 1)

+ ∑_{ℓ=1}^{p_{−}−1} (βρ^{+}(g_{p_{−}−ℓ,−}^{+} − g_{p_{−}−ℓ,−} − α^{+}h_{p_{−}−ℓ,−} − β^{+}f_{p_{−}−ℓ,−}^{+})

+ ρ^{+}(β(g_{p_{−}−ℓ,−} + g_{p_{−}−ℓ,−}^{−}) + h_{p_{−}−ℓ,−}^{−} − h_{p_{−}−1−ℓ,−}))(L^{−ℓ}f)(n + 1)

+ ( ∑_{ℓ=1}^{p_{−}−1} (g_{p_{−}−1−ℓ,−} − g_{p_{−}−1−ℓ,−}^{−} − αh_{p_{−}−1−ℓ,−}^{−})

+ ∑_{ℓ=1}^{p_{−}} β(α^{+}(g_{p_{−}−ℓ,−}^{+} + g_{p_{−}−ℓ,−}) − f_{p_{−}−ℓ,−}^{+})

+ ∑_{ℓ=1}^{p_{−}} βα^{+}(g_{p_{−}−ℓ,−} − g_{p_{−}−ℓ,−}^{+} + α^{+}h_{p_{−}−ℓ,−} + β^{+}f_{p_{−}−ℓ,−}^{+})

+ ∑_{ℓ=1}^{p_{−}−1} α^{+}(β(−g_{p_{−}−ℓ,−} − g_{p_{−}−ℓ,−}^{−}) − h_{p_{−}−ℓ,−}^{−} + h_{p_{−}−1−ℓ,−}))(L^{−ℓ}f)(n)

+ ρ^{−}ρ(g_{0,+}^{−} − g_{0,+}^{−−})(L^{p_{+}}f)(n − 2) − ρ^{−}ρ(α^{−}h_{p_{+}−1,+}^{−−} + β^{−}f_{p_{+}−1,+}^{−})f(n − 2) + ρ(2β^{−}g_{0,+}^{−} − h_{0,+}^{−})(L^{p_{+}}f)(n − 1) + ρ((ρ^{−})^{2}h_{p_{+}−1,+}^{−−} − (β^{−})^{2}f_{p_{+}−1,+}^{−})f(n − 1) + ρ^{+}(β(g_{0,+}^{+} + g_{0,+}^{−}) − h_{0,+})(L^{p_{+}}f)(n + 1) − ρ^{+}(β(α^{+}h_{p_{+}−1,+} + β^{+}f_{p_{+}−1,+}^{+}) − h_{p_{+}−1,+}^{−})f(n + 1) + (g_{0,+} − g_{0,+}^{−})(L^{p_{+}+1}f)(n) + (g_{1,+} − g_{1,+}^{−} − αh_{0,+}^{−} + βα^{+}(g_{0,+} − g_{0,+}^{−}) + α^{+}h_{0,+})(L^{p_{+}}f)(n) + (g_{p_{−}−1,−} − g_{p_{−}−1,−}^{−} − αh_{p_{−}−1,−}^{−} − βf_{p_{−}−1,−})f(n) + β(α^{+}(g_{p_{+},+}^{+} − g_{p_{+},+}) + f_{p_{−}−1,−} − f_{p_{+}−1,+}^{+})f(n) + α^{+}(β(βf_{p_{+}−1,+} + αh_{p_{+}−1,+}^{−}) + h_{p_{−}−1,−} − h_{p_{+}−1,+}^{−})f(n) − (βα^{+}(g_{0,−} + g_{0,−}^{−}) − βf_{−1,−} + α^{+}h_{0,−}^{−})(L^{−p_{−}}f)(n) − (αh_{p_{+}−1,+}^{−} + βf_{p_{+}−1,+})(βρ^{+}f(n + 1) + β^{−}ρf(n − 1) + ρ^{−}ρf(n − 2)) + (β^{−}ρ(g_{0,−}^{−} + g_{0,−}^{−} − β^{−}f_{0,−}^{−}) + (ρ^{−})^{2}ρh_{0,−}^{−−})(L^{−p_{−}}f)(n − 1)

20 F. GESZTESY, H. HOLDEN, J. MICHOR, AND G. TESCHL

+ (βρ^{+}(g_{0,−}^{+} + g_{0,−}^{−} − β^{+}f_{0,−}^{+} − α^{+}h_{0,−}) + ρ^{+}h_{0,−}^{−})(L^{−p_{−}}f)(n + 1) − ρh_{p_{−}−1,−}^{−}f(n − 1) − ρ^{+}h_{p_{−}−1,−}f(n + 1)

+ \frac{1}{2}(βρ^{+}(g_{p_{−},−}^{+} + g_{p_{+},+}^{+} + g_{p_{−},−} + g_{p_{+},+})f(n + 1)

+ β^{−}ρ(g_{p_{−},−}^{−} + g_{p_{+},+}^{−} + g_{p_{−},−} + g_{p_{+},+})f(n − 1)

− ρ^{−}ρ(g_{p_{−},−}^{−−} + g_{p_{+},+}^{−−} − g_{p_{−},−} − g_{p_{+},+})f(n − 2))

= \frac{ρ^{−}ρ}{2}(g_{p_{+},+}^{−−} − g_{p_{−},−}^{−−} + g_{p_{−},−} − g_{p_{+},+})f(n − 2)

+ (ρ(β^{−}(g_{p_{+},+}^{−−} + g_{p_{−},−}^{−}) − h_{p_{−}−1,−}^{−} + h_{p_{+}−1,+}^{−−})

+ \frac{β^{−}ρ}{2}(g_{p_{+},+}^{−} + g_{p_{−},−} − g_{p_{+},+} − g_{p_{−},−}^{−}))f(n − 1)

+ (β(α^{+}(g_{p_{+},+}^{+} + g_{p_{−},−}) + f_{p_{−}−1,−} − f_{p_{+}−1,+}^{+})

− α^{+}(β(g_{p_{+},+}^{−} + g_{p_{−},−}) − h_{p_{−}−1,−} + h_{p_{+}−1,+}^{−}))f(n) + (ρ^{+}(β(g_{p_{+},+}^{−} + g_{p_{−},−}) − h_{p_{−}−1,−} + h_{p_{+}−1,+}^{−})

+ \frac{βρ^{+}}{2} (g_{p_{+},+} + g_{p_{−},−}^{+} − g_{p_{+},+}^{+} − g_{p_{−},−}))f(n + 1),  (3.43)

where we also used (2.25).

Comparing coefficients finally shows that (3.38) is equivalent to

(ρ^{−}ρ)_{t\underline{p}} = ρ^{−}ρ(C^{−} + C),  (3.44)
(αρ^{−})_{t\underline{p}} = ρ^{−}A + αρ^{−}C^{−},  (3.45)
(βρ^{+})_{t\underline{p}} = ρ^{+}B + βρ^{+}C^{+},  (3.46)
(α^{+}β)_{t\underline{p}} = βA^{+} + α^{+}B,  (3.47)

where

A = i(α(g_{p_{+},+} + g_{p_{−},−}^{−}) − f_{p_{+}−1,+} + f_{p_{−}−1,−}^{−}),  (3.48)
B = i( − β(g_{p_{+},+}^{−} + g_{p_{−},−}) + h_{p_{−}−1,−} − h_{p_{+}−1,+}^{−}),  (3.49)
C = \frac{i}{2}(g_{p_{+},+} + g_{p_{−},−}^{−} − g_{p_{+},+}^{−} − g_{p_{−},−}).  (3.50)

In particular, (2.51) implies (3.38) since, by (2.53),

ρ_{t\underline{p}} = \frac{i}{2}ρ(g_{p_{+},+} + g_{p_{−},−}^{−} − g_{p_{+},+}^{−} − g_{p_{−},−}).  (3.51)

To prove the converse assertion (i.e., that (3.38) implies (2.51)), we argue as follows: Rewriting (3.45) and (3.46) using ρ = γ^{1/2} = (1 − αβ)^{1/2} and (3.44) yields

\begin{aligned}(1 + \frac{αβ}{2γ})α_{t\underline{p}} + \frac{α^{2}}{2γ} β_{t\underline{p}} & = A − αC, \\ \frac{β^{2}}{2γ} α_{t\underline{p}} + (1 + \frac{αβ}{2γ})β_{t\underline{p}} & = B − βC. \end{aligned}  (3.52)

This linear system is uniquely solvable since its determinant equals γ^{−1} and the solution reads

\begin{aligned}α_{t\underline{p}} & = A − \frac{α}{2}(βA + αB + 2γC), \\ β_{t\underline{p}} & = B − \frac{β}{2} (βA + αB + 2γC). \end{aligned}  (3.53)

Using (2.6) and (2.10) it is straightforward to check that βA+αB +2γC = 0 which shows that the converse assertion also holds. □

The Ablowitz–Ladik Lax pair in the special defocusing case, where β = \overline{α}, in the finite-dimensional context, was recently discussed by Nenciu [38].

4. Green’s functions and high- and low-energy expansions

In this section we discuss the Green’s function of an ℓ^{2}(ℤ)-realization of the difference expression L and systematically derive high- and low-energy expansions of solutions of an associated Riccati-type equation.

Throughout this section we make the following strengthened assumptions on the coefficients α and β.

Hypothesis 4.1. Suppose that α, β satisfy

α, β ∈ ℓ^{∞}(ℤ), α(n)β(n) ∉ {0, 1}, n ∈ ℤ.  (4.1)

Given Hypothesis 4.1 we introduce the ℓ^{2}(ℤ)-realization L̆ of the difference expression L in (3.5) by

L̆f = Lf, f ∈ dom(L̆) = ℓ^{2}(ℤ),  (4.2)

and similarly introduce the ℓ^{2}(ℤ)-realizations of the difference expression D, E, D^{−1}, and E^{−1} in (3.17)–(3.20) by

D̆f = Df, f ∈ dom(D̆) = ℓ^{2}(ℤ),  (4.3)
Ĕf = Ef, f ∈ dom(Ĕ) = ℓ^{2}(ℤ),  (4.4)
D̆^{−1}f = D^{−1}f, f ∈ dom(D̆^{−1}) = ℓ^{2}(ℤ),  (4.5)
Ĕ^{−1}f = E^{−1}f, f ∈ dom(Ĕ^{−1}) = ℓ^{2}(ℤ).  (4.6)

The following elementary result shows that these ℓ^{2}(ℤ)-realizations are meaningful; it will be used in the proof of Lemma 4.3 below.

Lemma 4.2. Assume Hypothesis 4.1. Then the operators D̆, D̆^{−1}, Ĕ, Ĕ^{−1}, L̆, and L̆^{−1} are bounded on ℓ^{2}(ℤ). In addition, (L̆ − z)^{−1} is norm analytic with respect to z in an open neighborhood of z = 0, and (L̆−z)^{−1} = −z^{−1}(I −z^{−1}L̆)^{−1} is analytic with respect to 1/z in an open neighborhood of 1/z = 0.

Proof. By Hypothesis 4.1, ρ^{2} = 1 − αβ, and (3.17)–(3.20), one infers that D̆, Ĕ, D̆^{−1}, Ĕ^{−1} are bounded operators on ℓ^{2}(ℤ) whose norms are bounded by

∥D̆∥, ∥Ĕ∥, ∥D̆^{−1}∥, ∥Ĕ^{−1}∥ ≤ 2∥ρ∥_{∞} + ∥α∥_{∞} + ∥β∥_{∞}
≤ 2(1 + ∥α∥_{∞} + ∥β∥_{∞}).  (4.7)

Since by (3.16),

L̆ = D̆Ĕ, L̆^{−1} = Ĕ^{−1}D̆^{−1},  (4.8)

the assertions of Lemma 4.2 are evident (alternatively, one can of course invoke (3.5) and (3.12)). □

To introduce the Green’s function of L̆, we need to digress a bit. Introducing the transfer matrix T (z, ·) associated with L by

T (z, n) = \begin{cases}ρ(n)^{−1} (\begin{matrix}α(n) & z \\ z^{−1} & β(n) \end{matrix}) , & n odd, \\ ρ(n)^{−1} (\begin{matrix}β(n) & 1 \\ 1 & α(n) \end{matrix}) , & n even, \end{cases} z ∈ ℂ \ \{0\}, n ∈ ℤ,  (4.9)

recalling that ρ = γ^{1/2} = (1 − αβ)^{1/2}, one then verifies that (cf. (2.27))

T (z, n) = A(z, n)z^{−1/2}ρ(n)^{−1}U(z, n)A(z, n − 1)^{−1}, z ∈ ℂ \ {0}, n ∈ ℤ. (4.10)

Here we introduced

A(z, n) = \begin{cases}(\begin{matrix}z^{1/2} & 0 \\ 0 & z^{−1/2} \end{matrix}) , & n odd, \\ (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) , & n even, \end{cases} z ∈ ℂ \ \{0\}, n ∈ ℤ.  (4.11)

Next, we consider a fundamental system of solutions

Ψ_{±}(z, ·) = (\begin{matrix}ψ_{1,±}(z, ·) \\ ψ_{2,±}(z, ·) \end{matrix})  (4.12)

of

U(z)Ψ_{±}^{−}(z) = Ψ_{±}(z), z ∈ ℂ \ (spec(L̆) ∪ {0}),  (4.13)

with spec(L̆) denoting the spectrum of L̆ and U given by (2.27), such that

det(Ψ_{−}(z), Ψ_{+}(z)) ≠ 0.  (4.14)

The precise form of Ψ_{±} will be chosen as a consequence of (4.20) below. Introducing in addition,

\begin{matrix}(\begin{matrix}u_{±}(z, n) \\ v_{±}(z, n) \end{matrix}) = C_{±}z^{−n/2}( ∏_{n′=1}^{n} ρ(n′)^{−1})A(z, n) (\begin{matrix}ψ_{1,±}(z, n) \\ ψ_{2,±}(z, n) \end{matrix}) , \\ z ∈ ℂ \ (spec(L̆) ∪ \{0\}), n ∈ ℤ, \end{matrix}  (4.15)

for some constants C_{±} ∈ ℂ \ {0}, (4.10) and (4.15) yield

T (z) (\begin{matrix}u_{±}^{−}(z) \\ v_{±}^{−}(z) \end{matrix}) = (\begin{matrix}u_{±}(z) \\ v_{±}(z) \end{matrix}) .  (4.16)

Moreover, one can show (cf. [28]) that

Lu_{±}(z) = zu_{±}(z), L^{⊤}v_{±}(z) = zv_{±}(z),  (4.17)
Dv_{±}(z) = u_{±}(z), Eu_{±}(z) = zv_{±}(z),  (4.18)

where

L = DE, L^{⊤} = ED,  (4.19)

and hence L^{⊤} represents the difference expression associated with the transpose of the infinite matrix L (cf. (3.3)) in the standard basis of ℓ^{2}(ℤ). Next, we choose Ψ_{±}(z) in such a manner (cf. again [28]), such that for all n_{0} ∈ ℤ,

(\begin{matrix}u_{±}(z, ·) \\ v_{±}(z, ·) \end{matrix}) ∈ ℓ^{2}([n_{0}, ±∞) ∩ ℤ)^{2}, z ∈ ℂ \ (spec(L̆) ∪ \{0\}).  (4.20)

Since by hypothesis z ∈ ℂ\spec(L̆), (u_{+}(z, ·), v_{+}(z, ·))^{⊤} and (u_{−}(z, ·), v_{−}(z, ·))^{⊤} are linearly independent since otherwise z would be an eigenvalue of L̆. This is of course consistent with (4.14) and (4.15).

The Green’s function of L̆, the ℓ^{2}(ℤ)-realization of the Lax difference expression L, is then of the form

G(z, n, n′) = (δ_{n}, (L̆ − z)^{−1}δ_{n′})  (4.21)

= \frac{−1}{z det (\begin{matrix}u_{+}(z, 0) & u_{−}(z, 0) \\ v_{+}(z, 0) & v_{−}(z, 0) \end{matrix})}

× \begin{cases}v_{−}(z, n′)u_{+}(z, n), & n′ < n or n = n′ even, \\ v_{+}(z, n′)u_{−}(z, n), & n′ > n or n = n′ odd, \end{cases} n, n′ ∈ ℤ,

= −\frac{1}{4z} \frac{(1 − ϕ_{+}(z, 0))(1 − ϕ_{−}(z, 0))}{ϕ_{+}(z, 0) − ϕ_{−}(z, 0)}

× \begin{cases}v_{−}(z, n′)u_{+}(z, n), & n′ < n or n = n′ even, \\ v_{+}(z, n′)u_{−}(z, n), & n′ > n or n = n′ odd, \end{cases} n, n′ ∈ ℤ,  (4.22)
z ∈ ℂ \ (spec(L̆) ∪ {0}).

Introducing

ϕ_{±}(z, n) = \frac{ψ_{2,±}(z, n)}{ψ_{1,±}(z, n)}, z ∈ ℂ \ (spec(L̆) ∪ \{0\}), n ∈ ℕ,  (4.23)

then (4.13) implies that ϕ_{±} satisfy the Riccati-type equation

αϕ_{±}ϕ_{±}^{−} − ϕ_{±}^{−} + zϕ_{±} = zβ,  (4.24)

and one introduces in addition,

𝔣 = \frac{2}{ϕ_{+} − ϕ_{−}} ,  (4.25)
𝔤 = \frac{ϕ_{+} + ϕ_{−}}{ϕ_{+} − ϕ_{−}} ,  (4.26)
𝔥 = \frac{2ϕ_{+}ϕ_{−}}{ϕ_{+} − ϕ_{−}} .  (4.27)

Using the Riccati-type equation (4.24) and its consequences,

α(ϕ_{+}ϕ_{+}^{−} − ϕ_{−}ϕ_{−}^{−}) − (ϕ_{+}^{−} − ϕ_{−}^{−}) + z(ϕ_{+} − ϕ_{−}) = 0,  (4.28)
α(ϕ_{+}ϕ_{+}^{−} + ϕ_{−}ϕ_{−}^{−}) − (ϕ_{+}^{−} + ϕ_{−}^{−}) + z(ϕ_{+} + ϕ_{−}) = 2zβ,  (4.29)

one then derives the identities

z(𝔤^{−} − 𝔤) + zβ𝔣 + α𝔥^{−} = 0,  (4.30)
zβ𝔣^{−} + α𝔥 − 𝔤 + 𝔤^{−} = 0,  (4.31)

−𝔣 + z𝔣^{−} + α(𝔤 + 𝔤^{−}) = 0,  (4.32)
zβ(𝔤^{−} + 𝔤) − z𝔥 + 𝔥^{−} = 0,  (4.33)
𝔤^{2} − 𝔣𝔥 = 1.  (4.34)

For the connection between 𝔣, 𝔤, and 𝔥 and the Green’s function of L one finally obtains

𝔣(z, n) = −2α(n)(zG(z, n, n) + 1) − 2ρ(n)z \begin{cases}G(z, n − 1, n), & n even, \\ G(z, n, n − 1), & n odd, \end{cases}
𝔤(z, n) = −2zG(z, n, n) − 1,  (4.35)
𝔥(z, n) = −2β(n)zG(z, n, n) − 2ρ(n)z \begin{cases}G(z, n, n − 1), & n even, \\ G(z, n − 1, n), & n odd, \end{cases}

illustrating the spectral theoretic content of 𝔣, 𝔤, and 𝔥.

We are particularly interested in the asymptotic expansion of ϕ_{±} in a neighborhood of the points z = 0 and 1/z = 0 and turn to this topic next.

Lemma 4.3. Assume that α, β satisfy Hypothesis 4.1. Then ϕ_{±} have the following convergent expansions with respect to 1/z around 1/z = 0 and with respect to z around z = 0,

ϕ_{±}(z) =_{z→∞} \begin{cases}∑_{j=0}^{∞} ϕ_{j,+}^{∞}z^{−j}, \\ ∑_{j=−1}^{∞} ϕ_{j,−}^{∞}z^{−j}, \end{cases}  (4.36)

ϕ_{±}(z) =_{z→0} \begin{cases}∑_{j=0}^{∞} ϕ_{j,+}^{0}z^{j}, \\ ∑_{j=1}^{∞} ϕ_{j,−}^{0}z^{j}, \end{cases}  (4.37)

where

ϕ_{0,+}^{∞} = β, ϕ_{1,+}^{∞} = β^{−}γ,
ϕ_{j+1,+}^{∞} = (ϕ_{j,+}^{∞})^{−} − α ∑_{ℓ=0}^{j}(ϕ_{j−ℓ,+}^{∞})^{−}ϕ_{ℓ,+}^{∞}, j ∈ ℕ,  (4.38)

ϕ_{−1,−}^{∞} = −\frac{1}{α^{+}} , ϕ_{0,−}^{∞} = \frac{α^{++}}{(α^{+})^{2}} γ^{+},

ϕ_{j+1,−}^{∞} = −\frac{α^{++}}{α^{+}} ϕ_{j,−}^{∞} + α^{++} ∑_{ℓ=0}^{j} ϕ_{j−ℓ,−}^{∞}(ϕ_{ℓ,−}^{∞})^{+}, j ∈ ℕ_{0},  (4.39)

ϕ_{0,+}^{0} = \frac{1}{α}, ϕ_{1,+}^{0} = −\frac{α^{−}}{α^{2}} γ,
ϕ_{j+1,+}^{0} = (ϕ_{j,+}^{0})^{+} + α^{+} ∑_{ℓ=0}^{j+1}(ϕ_{j+1−ℓ,+}^{0})^{+}ϕ_{ℓ,+}^{0}, j ∈ ℕ,  (4.40)
ϕ_{1,−}^{0} = −β^{+},
ϕ_{j+1,−}^{0} = (ϕ_{j,−}^{0})^{+} + α^{+} ∑_{ℓ=1}^{j} ϕ_{j+1−ℓ,−}^{0}(ϕ_{ℓ,−}^{0})^{+}, j ∈ ℕ.  (4.41)

Proof. Since

ϕ_{±} = \frac{𝔤 ± 1}{𝔣},  (4.42)

combining Lemma 4.2, (4.21) and (4.35) proves that ϕ_{±} has a convergent expansion with respect to z and 1/z in a neighborhood of z = 0 and 1/z = 0, respectively.

The explicit expansion coefficients ϕ_{j,±}^{∞} are then readily derived by making the ansatz

ϕ_{±} =_{z→∞} ∑_{j=−1}^{∞} ϕ_{j,±}^{∞}z^{−j}.  (4.43)

Inserting (4.43) into the Riccati-type equation (4.24) one finds

0 = αϕ_{±}ϕ_{±}^{−} − ϕ_{±}^{−} + z(ϕ_{±} − β) = (αϕ_{−1,±}ϕ_{−1,±}^{−} + ϕ_{−1,±})z^{2} + O(z^{2}), (4.44)

which yields the case distinction above and the formulas for ϕ_{j,±}^{∞}. The corresponding expansion coefficients ϕ_{j,±}^{0} are obtained analogously by making the ansatz

ϕ_{±} =_{z→0} ∑_{j=0}^{∞} ϕ_{j,±}^{0}z^{j}. □

For the record we list a few explicit expressions:

ϕ_{0,+}^{∞} = β,
ϕ_{1,+}^{∞} = β^{−}γ,
ϕ_{2,+}^{∞} = γ( − α(β^{−})^{2} + β^{−−}γ^{−}),
ϕ_{−1,−}^{∞} = −\frac{1}{α^{+}} ,
ϕ_{0,−}^{∞} = \frac{α^{++}}{(α^{+})^{2}} γ^{+},
ϕ_{1,−}^{∞} = \frac{γ^{+}}{(α^{+})^{3}} (α^{+}α^{+++}γ^{++} − (α^{++})^{2}),
ϕ_{0,+}^{0} = \frac{1}{α},
ϕ_{1,+}^{0} = −\frac{α^{−}}{α^{2}} γ,
ϕ_{2,+}^{0} = \frac{γ}{α^{3}} ((α^{−})^{2} − α^{−−}αγ^{−}),
ϕ_{1,−}^{0} = −β^{+},
ϕ_{2,−}^{0} = −γ^{+}β^{++},
ϕ_{3,−}^{0} = γ^{+}(α^{+}(β^{++})^{2} − γ^{++}β^{+++}),
ϕ_{4,−}^{0} = γ^{+}( − (α^{+})^{2}(β^{++})^{3}
+ γ^{++}(2α^{+}β^{++}β^{+++} + α^{++}(β^{+++})^{2} − γ^{+++}β^{++++})), etc.

Later on we will also need the convergent expansions of ln(z + α^{+}ϕ_{±}(z)) with respect to z and 1/z. We will separately provide all four expansions of ln(z + α^{+}ϕ_{±}(z)) around 1/z = 0 and z = 0 and repeatedly use the general formula

ln (1 + ∑_{j=1}^{∞} ω_{j}z^{±j}) = ∑_{j=1}^{∞} σ_{j}z^{±j},  (4.45)

26 F. GESZTESY, H. HOLDEN, J. MICHOR, AND G. TESCHL

where

σ_{1} = ω_{1}, σ_{j} = ω_{j} − ∑_{ℓ=1}^{j−1} \frac{ℓ}{j} ω_{j−ℓ}σ_{ℓ}, j ≥ 2, (4.46)

and |z| as |z| → 0, respectively, 1/|z| as |z| → ∞, are assumed to be sufficiently small in (4.45). We start by expanding ϕ_{+} around 1/z = 0

ln(z + α^{+}ϕ_{+}(z)) = ln (z + α^{+} ∑_{j=0}^{∞} ϕ_{j,+}^{∞}z^{−j})

= ln (1 + α^{+} ∑_{j=0}^{∞} ϕ_{j,+}^{∞}z^{−j−1})

= ln (1 + α^{+} ∑_{j=1}^{∞} ϕ_{j−1,+}^{∞}z^{−j})

= ∑_{j=1}^{∞} ρ_{j,+}^{∞}z^{−j},  (4.47)

where

ρ_{1,+}^{∞} = α^{+}ϕ_{0,+}^{∞}, ρ_{j,+}^{∞} = α^{+}(ϕ_{j−1,+}^{∞} − ∑_{ℓ=1}^{j−1} \frac{ℓ}{j} ϕ_{j−1−ℓ,+}^{∞}ρ_{ℓ,+}^{∞}), j ≥ 2. (4.48)

An expansion of ϕ_{−} around 1/z = 0 yields

ln(z + α^{+}ϕ_{−}(z)) = ln (z + α^{+} ∑_{j=−1}^{∞} ϕ_{j,−}^{∞}z^{−j})

= ln (\frac{α^{++}γ^{+}}{α^{+}} ) + ln (1 + \frac{(α^{+})^{2}}{α^{++}γ^{+}} ∑_{j=1}^{∞} ϕ_{j,−}^{∞}z^{−j})

= ln (\frac{α^{++}}{α^{+}} ) + ln γ^{+} + ln (1 + \frac{(α^{+})^{2}}{α^{++}γ^{+}} ∑_{j=1}^{∞} ϕ_{j,−}^{∞}z^{−j})

= ln (\frac{α^{++}}{α^{+}} ) + ln γ^{+} + ∑_{j=1}^{∞} ρ_{j,−}^{∞}z^{−j},  (4.49)

where

ρ_{1,−}^{∞} = \frac{(α^{+})^{2}}{α^{++}γ^{+}} ϕ_{1,−}^{∞}, ρ_{j,−}^{∞} = \frac{(α^{+})^{2}}{α^{++}γ^{+}} (ϕ_{j,−}^{∞} − ∑_{ℓ=1}^{j−1} \frac{ℓ}{j} ϕ_{j−ℓ,−}^{∞}ρ_{ℓ,−}^{∞}), j ≥ 2. (4.50)

For the expansion of ϕ_{+} around z = 0 one gets

ln(z + α^{+}ϕ_{+}(z)) = ln (z + α^{+} ∑_{j=0}^{∞} ϕ_{j,+}^{0}z^{j})

= ln (\frac{α^{+}}{α} ) + ln (1 + \frac{α}{α^{+}} (1 + α^{+}ϕ_{1,+}^{0})z + α ∑_{j=2}^{∞} ϕ_{j,+}^{0}z^{j})

= ln (\frac{α^{+}}{α} ) + ∑_{j=1}^{∞} ρ_{j,+}^{0}z^{j},  (4.51)

where

\begin{aligned}ρ_{1,+}^{0} & = \frac{α}{α^{+}} (1 + α^{+}ϕ_{1,+}^{0}), ρ_{2,+}^{0} = αϕ_{2,+}^{0} − \frac{1}{2}(ρ_{1,+}^{0})^{2}, \\ ρ_{j,+}^{0} & = α(ϕ_{j,+}^{0} − ∑_{j=1}^{j−2} \frac{ℓ}{j} ϕ_{j−ℓ,+}^{0}ρ_{ℓ,+}^{0}) − \frac{j − 1}{j}ρ_{1,+}^{0}ρ_{j−1,+}^{0}, j ≥ 3. \end{aligned}  (4.52)

Finally, the expansion of ϕ_{−} around z = 0 is given by

ln(z + α^{+}ϕ_{−}(z)) = ln (z + α^{+} ∑_{j=1}^{∞} ϕ_{j,−}^{0}z^{j})

= ln (γ^{+}z + α^{+} ∑_{j=2}^{∞} ϕ_{j,−}^{0}z^{j})

= ln z + ln γ^{+} + ln (1 + \frac{α^{+}}{γ^{+}} ∑_{j=1}^{∞} ϕ_{j+1,−}^{0}z^{j})

= ln z + ln γ^{+} + ∑_{j=1}^{∞} ρ_{j,−}^{0}z^{j},  (4.53)

where

ρ_{1,−}^{0} = \frac{α^{+}}{γ^{+}} ϕ_{2,−}^{0}, ρ_{j,−}^{0} = \frac{α^{+}}{γ^{+}} (ϕ_{j+1,−}^{0} − ∑_{ℓ=1}^{j−1} \frac{ℓ}{j} ϕ_{j+1−ℓ,−}^{0}ρ_{ℓ,−}^{0}), j ≥ 2. (4.54)

Explicitly, the first expansion coefficients are given by

\begin{aligned}ρ_{1,+}^{∞} & = α^{+}β, \\ ρ_{2,+}^{∞} & = −\frac{1}{2}(α^{+}β)^{2} + γα^{+}β^{−}, \\ ρ_{3,+}^{∞} & = \frac{1}{3}(α^{+}β)^{3} − γ(γ^{−}α^{+}β^{−−} − (α^{+})^{2}β^{−}β − αα^{+}(β^{−})^{2}), \\ ρ_{1,−}^{∞} & = −α^{+++}β^{++} + (S^{+} − I)\frac{α^{++}}{α^{+}} , \\ ρ_{1,+}^{0} & = α^{−}β + (S^{+} − I)\frac{α^{−}}{α} , \\ ρ_{1,−}^{0} & = −α^{+}β^{++}, \\ ρ_{2,−}^{0} & = \frac{1}{2}(α^{+}β^{++})^{2} − γ^{++}α^{+}β^{+++}, \\ ρ_{3,−}^{0} & = −\frac{1}{3}(α^{+}β^{++})^{3} \\ & + γ^{++}( − γ^{+++}α^{+}β^{++++} + (α^{+})^{2}β^{++}β^{+++} + α^{+}α^{++}(β^{+++})^{2}), etc. \end{aligned} (4.55)

The next result shows that ĝ_{j,+} and ±jρ_{j,±}^{∞}, respectively ĝ_{j,−} and ±jρ_{j,±}^{0}, are equal up to terms that are total differences, that is, are of the form (S^{+} − I)d_{j,±} for some sequence d_{j,±}. The exact form of d_{j,±} will not be needed later. In the proof we will heavily use the equations (4.30)–(4.34).

Lemma 4.4. Suppose Hypothesis 4.1 holds. Then

ĝ_{j,+} = −jρ_{j,+}^{∞} + (S^{+} − I)d_{j,+} = jρ_{j,−}^{∞} + (S^{+} − I)e_{j,+}, j ∈ ℕ,  (4.56)
ĝ_{j,−} = −jρ_{j,+}^{0} + (S^{+} − I)d_{j,−} = jρ_{j,−}^{0} + (S^{+} − I)e_{j,−}, j ∈ ℕ,  (4.57)

28 F. GESZTESY, H. HOLDEN, J. MICHOR, AND G. TESCHL

for some polynomials d_{j,±}, e_{j,±}, j ∈ ℕ, in α and β and certain shifts thereof. Proof. We consider the case for ĝ_{j,+} first. Our aim is to show that

\frac{d}{dz} ln(z + α^{+}ϕ_{+}) = −\frac{1}{2z} 𝔤 + \frac{1}{2z} + (S^{+} − I)K + (S^{+} − I)M,  (4.58)

where

K = \frac{1}{2}(𝔤\frac{𝔣̇}{𝔣} − 𝔤̇), M = \frac{1}{2} \frac{𝔣̇}{𝔣},  (4.59)

which implies (4.56) by (4.47). Here · denotes d/dz.

Since ϕ_{+} = (𝔤 + 1)/𝔣,

\frac{d}{dz} ln(z + α^{+}ϕ_{+}) = \frac{1 + α^{+}ϕ̇_{+}}{z + α^{+}ϕ_{+}} = \frac{𝔣^{2} + α^{+}(𝔣𝔤̇ − 𝔣̇𝔤 − 𝔣̇)}{𝔣(z𝔣 + α^{+}𝔤 + α^{+})} \frac{z𝔣 + α^{+}𝔤 − α^{+}}{z𝔣 + α^{+}𝔤 − α^{+}} . (4.60)

Next we treat the denominator of (4.60) using (4.30), (4.32),

𝔣((z𝔣 + α^{+}𝔤)^{2} − (α^{+})^{2}) = 𝔣((z𝔣 + α^{+}𝔤)^{2} − (α^{+})^{2}(𝔤^{2} − 𝔣𝔥))

= z𝔣^{2}(z𝔣 + α^{+}𝔤 + α^{+}𝔤 + (α^{+})^{2} \frac{1}{z}𝔥)

= z𝔣^{2}(𝔣^{+} − α^{+}𝔤^{+} + α^{+}𝔤^{+} − α^{+}β^{+}𝔣^{+})

= zγ^{+}𝔣^{2}𝔣^{+}.  (4.61)

Expanding the numerator in (4.60) and applying (4.30), (4.32), and their derivatives with respect to z as well as 2𝔤𝔤̇ = 𝔣̇𝔥 + 𝔣𝔥̇ yields

(z𝔣 + α^{+}𝔤 − α^{+})(𝔣^{2} + α^{+}(𝔣𝔤̇ − 𝔣̇𝔤 − 𝔣̇))

= 𝔣(z𝔣^{2} + zα^{+}(𝔣𝔤̇ − 𝔣̇𝔤) + α^{+}𝔣𝔤 + \frac{1}{2}(α^{+})^{2}(𝔣𝔥̇ − 𝔣̇𝔥) − α^{+}(𝔣 + z𝔣̇ + α^{+}𝔤̇))

= \frac{𝔣}{2}(2z𝔣^{2} + zα^{+}𝔣𝔤̇ + z𝔣(−α^{+}𝔤̇^{+} − z𝔣̇ − 𝔣 + 𝔣̇^{+}) − zα^{+}𝔣̇𝔤 + z𝔣̇(α^{+}𝔤^{+} + z𝔣 − 𝔣^{+})

+ α^{+}𝔣𝔤 + 𝔣(−α^{+}𝔤^{+} − z𝔣 + 𝔣^{+}) + α^{+}𝔣(−β^{+}𝔣^{+} − zβ^{+}𝔣̇^{+} − 𝔤 + 𝔤^{+} − z𝔤̇ + z𝔤̇^{+}) + α^{+}𝔣̇(zβ^{+}𝔣^{+} + z𝔤 − z𝔤^{+}) − 2α^{+}(𝔣 + z𝔣̇ + α^{+}𝔤̇))

= \frac{𝔣}{2}(γ^{+}𝔣𝔣^{+} + zγ^{+}𝔣𝔣̇^{+} − zγ^{+}𝔣̇𝔣^{+} − 2α^{+}(𝔣 + z𝔣̇ + α^{+}𝔤̇)).  (4.62)

In summary,

\frac{d}{dz} ln(z + α^{+}ϕ_{+}) = \frac{1}{2z} + (S^{+} − I)M − \frac{α^{+}}{zγ^{+}𝔣𝔣^{+}} (𝔣 + z𝔣̇ + α^{+}𝔤̇).  (4.63)

We multiply the numerator on the right-hand side by −2 = −2(𝔤^{2} − 𝔣𝔥) and use again (4.30), (4.32), and their derivatives:

2α^{+}(𝔣𝔥 − 𝔤^{2})(𝔣 + z𝔣̇ + α^{+}𝔤̇)

= 2α^{+}𝔣𝔥(𝔣 + z𝔣̇ + α^{+}𝔤̇) − 2α^{+}𝔣𝔤^{2} − 2zα^{+}𝔣̇𝔤^{2} − (α^{+})^{2}𝔤(𝔣̇𝔥 + 𝔣𝔥̇)

= α^{+}𝔣𝔥(𝔣 + z𝔣̇) + α^{+}𝔣𝔤̇(−zβ^{+}𝔣^{+} − z𝔤 + z𝔤^{+})

+ 𝔣(−zβ^{+}𝔣^{+} − z𝔤 + z𝔤^{+})(𝔣̇^{+} − α^{+}𝔤̇^{+})

− α^{+}𝔣𝔤^{2} + 𝔣𝔤(α^{+}𝔤^{+} + z𝔣 − 𝔣^{+}) − zα^{+}𝔣̇𝔤^{2} + z𝔣̇𝔤(α^{+}𝔤^{+} + z𝔣 − 𝔣^{+})

+ α^{+}𝔣̇𝔤(zβ^{+}𝔣^{+} + z𝔤 − z𝔤^{+}) + α^{+}𝔣𝔤(β^{+}𝔣^{+} + zβ^{+}𝔣̇^{+} + 𝔤 − 𝔤^{+} + z𝔤̇ − z𝔤̇^{+}) = α^{+}𝔣𝔥(𝔣 + z𝔣̇) + α^{+}𝔣𝔤̇(−zβ^{+}𝔣^{+} − z𝔤 + z𝔤^{+}) + 𝔣𝔣̇^{+}(−zβ^{+}𝔣^{+} − z𝔤 + z𝔤^{+})

CONSERVATION LAWS AND HAMILTONIANS FOR THE AL HIERARCHY 29

+ α^{+}𝔣𝔤̇^{+}(zβ^{+}𝔣^{+} + z𝔤 − z𝔤^{+}) + z𝔣^{2}𝔤 − γ^{+}𝔣𝔣^{+}𝔤 + z^{2}𝔣𝔣̇𝔤 − zγ^{+}𝔣̇𝔣^{+}𝔤 + zβ^{+}𝔣𝔣̇^{+}(−α^{+}𝔤^{+} − z𝔣 + 𝔣^{+}) + zα^{+}𝔣𝔤(𝔤̇ − 𝔤̇^{+})

= α^{+}𝔣𝔥(𝔣 + z𝔣̇) − zα^{+}β^{+}𝔣𝔣^{+}(𝔤̇ − 𝔤̇^{+}) − z𝔣(α^{+}𝔤 + z𝔣 − 𝔣^{+})(𝔤̇ − 𝔤̇^{+})

− z𝔣𝔣̇^{+}𝔤 + z𝔣^{2}𝔤 − γ^{+}𝔣𝔣^{+}𝔤 + z^{2}𝔣𝔣̇𝔤 − zγ^{+}𝔣̇𝔣^{+}𝔤 + zγ^{+}𝔣𝔣̇^{+}𝔤^{+} − z^{2}β^{+}𝔣^{2}𝔣̇^{+} = zα^{+}𝔣𝔣̇𝔥 − z𝔣^{2}(zβ^{+}𝔣̇^{+} + z𝔤̇ − z𝔤̇^{+} − \frac{1}{z}α^{+}𝔥) + z𝔣𝔤(𝔣 + z𝔣̇ − α^{+}𝔤̇ − 𝔣̇^{+} + α^{+}𝔤̇^{+})

+ zγ^{+}𝔣𝔣^{+}(𝔤̇ − 𝔤̇^{+}) − γ^{+}𝔣𝔣^{+}𝔤 − zγ^{+}𝔣̇𝔣^{+}𝔤 + zγ^{+}𝔣𝔣̇^{+}𝔤^{+} = zα^{+}𝔣(𝔣̇𝔥 + 𝔣𝔥̇ − 2𝔤𝔤̇) + zγ^{+}𝔣𝔣^{+}(𝔤̇ − 𝔤̇^{+}) − γ^{+}𝔣𝔣^{+}𝔤 − zγ^{+}𝔣̇𝔣^{+}𝔤 + zγ^{+}𝔣𝔣̇^{+}𝔤^{+}

= zγ^{+}𝔣𝔣^{+}(𝔤̇ − 𝔤̇^{+}) − γ^{+}𝔣𝔣^{+}𝔤 − zγ^{+}𝔣̇𝔣^{+}𝔤 + zγ^{+}𝔣𝔣̇^{+}𝔤^{+}.  (4.64)

Inserting this in (4.63) finally yields (4.58). The result for ĝ_{j,−} is derived similarly starting from ϕ_{−} = (𝔤 − 1)/𝔣. □

5. Local conservation laws

Throughout this section (and with the only exception of Theorem 5.5) we make the following assumption: Hypothesis 5.1. Suppose that α, β: ℤ × ℝ → ℂ satisfy

\begin{matrix}sup_{(n,t\underline{p})∈ℤ×ℝ} (|α(n, t\underline{p})| + |β(n, t\underline{p})|) < ∞, \\ a(n, ·), b(n, ·) ∈ C^{1}(ℝ), n ∈ ℤ, α(n, t\underline{p})β(n, t\underline{p}) ∉ \{0, 1\}, (n, t\underline{p}) ∈ ℤ × ℝ. \end{matrix} (5.1)

In accordance with the notation introduced in (4.2)–(4.6) we denote the bounded difference operator defined on ℓ^{2}(ℤ), generated by the finite difference expression P\underline{p} in (3.26), by the symbol P̆\underline{p}. Similarly, the bounded finite difference operator in ℓ^{2}(ℤ) generated by P ⊤ ; \underline{p} in (3.33) is then denoted by P̆ ⊤ ; \underline{p}.

We start with the following existence result. Theorem 5.2. Assume Hypothesis 5.1 and suppose α, β satisfy AL\underline{p}(α, β) = 0 for some \underline{p} ∈ ℕ_{0}^{2}. In addition, let t\underline{p} ∈ ℝ and z ∈ ℂ \ (spec(L̆(t\underline{p})) ∪ \{0\}). Then there exist Weyl–Titchmarsh-type solutions u_{±} = u_{±}(z, n, t\underline{p}) and v_{±} = v_{±}(z, n, t\underline{p}) such that for all n_{0} ∈ ℤ,

(\begin{matrix}u_{±}(z, · , t\underline{p}) \\ v_{±}(z, · , t\underline{p}) \end{matrix}) ∈ ℓ^{2}([n_{0}, ±∞) ∩ ℤ)^{2}, u_{±}(z, n, ·), v_{±}(z, n, ·) ∈ C^{1}(ℝ), (5.2)

and u_{±} and v_{±} simultaneously satisfy the following equations in the weak sense

L̆(t\underline{p})u_{±}(z, · , t\underline{p}) = zu_{±}(z, · , t\underline{p}),  (5.3)
u_{±,t\underline{p}}(z, · , t\underline{p}) = P̆\underline{p}(t\underline{p})u_{±}(z, · , t\underline{p}),  (5.4)

and

L̆^{⊤}(t\underline{p})v_{±}(z, · , t\underline{p}) = zv_{±}(z, · , t\underline{p}),  (5.5)
v_{±,t\underline{p}}(z, · , t\underline{p}) = −P̆ ⊤ ; \underline{p}(t\underline{p})v_{±}(z, · , t\underline{p}),  (5.6)

respectively.

Proof. Applying (L̆(t\underline{p}) − zI)^{−1} to δ_{0} (cf. (4.21)) yields the existence of Weyl– Titchmarsh-type solutions ũ_{±} of Lu = zu satisfying (5.2). Next, using the Lax commutator equation (3.38) one computes

\begin{aligned}zũ_{±,t\underline{p}} & = (Lũ_{±})_{t\underline{p}} = L_{t\underline{p}} ũ_{±} + Lũ_{±,t\underline{p}} = [P\underline{p}, L]ũ_{±} + Lũ_{±,t\underline{p}} \\ & = zP\underline{p}ũ_{±} − LP\underline{p}ũ_{±} + Lũ_{±,t\underline{p}} \end{aligned}  (5.7)

and hence

(L − zI)(ũ_{±,t\underline{p}} − P\underline{p}ũ_{±}) = 0.  (5.8)

Thus, ũ_{±} satisfy

ũ_{±,t\underline{p}} − P\underline{p}ũ_{±} = C_{±}ũ_{±} + D_{±}ũ_{∓}.  (5.9)

Introducing ũ_{±} = c_{±}u_{±}, and choosing c_{±} such that c_{±,t\underline{p}} = C_{±}c_{±}, one obtains

u_{±,t\underline{p}} − P\underline{p}u_{±} = D_{±}u_{∓}.  (5.10)

Since u_{±} ∈ ℓ^{2}([n_{0}, ±∞) ∩ ℤ), n_{0} ∈ ℤ, and α, β satisfy Hypothesis 5.1, (3.29) shows that P\underline{p}u_{±} ∈ ℓ^{2}([n_{0}, ±∞) ∩ ℤ). Moreover, since

u_{±}(z, n, t\underline{p}) = d_{±}(t\underline{p})(L̆(t\underline{p}) − zI)^{−1}δ_{0})(n)  (5.11)

for n ∈ [±1, ∞) ∩ ℤ and some d_{±} ∈ C^{1}(ℝ), the calculation

u_{±,t\underline{p}} = d_{±,t\underline{p}}(L̆ − zI)^{−1}δ_{0} − d_{±}(L̆ − zI)^{−1}L̆_{t\underline{p}}(L̆ − zI)^{−1}δ_{0}  (5.12)

also yields u_{±,t\underline{p}} ∈ ℓ^{2}([n_{0}, ±∞) ∩ ℤ). But then D_{±} = 0 in (5.10) since u_{∓} ∉ ℓ^{2}([n_{0}, ±∞) ∩ ℤ). This proves (5.4).

Equations (5.2), (5.5), and (5.6) for v_{±} are proved similarly replacing L, P\underline{p} by L^{⊤}, P ⊤ ; \underline{p} and observing that (3.38) implies

L_{t\underline{p}}^{⊤}(t\underline{p}) + [P ⊤ ; \underline{p}(t\underline{p}), L^{⊤}(t\underline{p})] = 0, t\underline{p} ∈ ℝ.  (5.13)
□

For the remainder of this section we will always refer to the Weyl–Titchmarsh solutions u_{±}, v_{±} introduced in Theorem 5.2. Given u_{±}, v_{±}, we now introduce

Ψ_{±}(z, · , t\underline{p}) = (\begin{matrix}ψ_{1,±}(z, · , t\underline{p}) \\ ψ_{2,±}(z, · , t\underline{p}) \end{matrix}) , z ∈ ℂ \ (spec(L̆(t\underline{p})) ∪ \{0\}), t\underline{p} ∈ ℝ, (5.14)

by (cf. (4.15))

\begin{matrix}(\begin{matrix}ψ_{1,±}(z, n, t\underline{p}) \\ ψ_{2,±}(z, n, t\underline{p}) \end{matrix}) = D(t\underline{p})z^{n/2}( ∏_{n′=1}^{n} ρ(n′, t\underline{p}))A(z, n)^{−1} (\begin{matrix}u_{±}(z, n, t\underline{p}) \\ v_{±}(z, n, t\underline{p}) \end{matrix}) , \\ z ∈ ℂ \ (spec(L̆(t\underline{p})) ∪ \{0\}), (n, t\underline{p}) ∈ ℤ × ℝ, \end{matrix} (5.15)

with the choice of normalization

D(t\underline{p}) = exp (\frac{i}{2}(g_{p_{+},+}(0) − g_{p_{−},−}(0))t\underline{p})D(0), t\underline{p} ∈ ℝ,  (5.16)

for some constant D(0) ∈ ℂ \ {0}.

Lemma 5.3. Assume Hypothesis 5.1 and suppose α, β satisfy AL\underline{p}(α, β) = 0 for some \underline{p} ∈ ℕ_{0}^{2}. In addition, let t\underline{p} ∈ ℝ and z ∈ ℂ \ (spec(L̆(t\underline{p})) ∪ \{0\}). Then Ψ_{±}(z, · , t\underline{p}) defined in (5.15) satisfy

U(z, · , t\underline{p})Ψ_{±}^{−}(z, · , t\underline{p}) = Ψ_{±}(z, · , t\underline{p}),  (5.17)
Ψ_{±,t\underline{p}}(z, · , t\underline{p}) = V + ; \underline{p}(z, · , t\underline{p})Ψ_{±}(z, · , t\underline{p}).  (5.18)

In addition, Ψ_{−}(z, · , t\underline{p}) and Ψ_{+}(z, · , t\underline{p}) are linearly independent.

Proof. Equation (5.17) is equivalent to

(\begin{matrix}ψ_{1,±} \\ ψ_{2,±} \end{matrix}) = (\begin{matrix}zψ_{1,±}^{−} + αψ_{2,±}^{−} \\ zβψ_{1,±}^{−} + ψ_{2,±} \end{matrix}) .  (5.19)

Using (4.11) and (5.15) one obtains

(\begin{matrix}ψ_{1,±} \\ ψ_{2,±} \end{matrix}) = Dz^{n/2}( ∏_{n′=1}^{n} ρ(n′)) \begin{cases}(\begin{matrix}z^{−1/2}u_{±} \\ z^{1/2}v_{±} \end{matrix}) , & n odd, \\ (\begin{matrix}v_{±} \\ u_{±} \end{matrix}) , & n even. \end{cases}  (5.20)

Inserting (5.20) into (5.19), one finds that (5.19) is equivalent to (4.16), thereby proving (5.17).

Equation (5.18) is equivalent to

(\begin{matrix}ψ_{1,±,t\underline{p}} \\ ψ_{2,±,t\underline{p}} \end{matrix}) = i (\begin{matrix}G\underline{p}ψ_{1,±} − F\underline{p}ψ_{2,±} \\ H\underline{p}ψ_{1,±} − K\underline{p}ψ_{2,±} \end{matrix}) .  (5.21)

We first consider the case when n is odd. Using (5.20), the right-hand side of (5.21) reads,

i (\begin{matrix}G\underline{p}ψ_{1,±} − F\underline{p}ψ_{2,±} \\ H\underline{p}ψ_{1,±} − K\underline{p}ψ_{2,±} \end{matrix}) = iDz^{(n−1)/2}( ∏_{n′=1}^{n} ρ(n′)) (\begin{matrix}G\underline{p}u_{±} − zF\underline{p}v_{±} \\ H\underline{p}u_{±} − zK\underline{p}v_{±} \end{matrix}) . (5.22)

Equation (5.20) then implies

(\begin{matrix}ψ_{1,±,t\underline{p}} \\ ψ_{2,±,t\underline{p}} \end{matrix}) = D_{t\underline{p}}z^{n/2}( ∏_{n′=1}^{n} ρ(n′)) (\begin{matrix}z^{−1/2}u_{±} \\ z^{1/2}v_{±} \end{matrix})  (5.23)

+ Dz^{n/2}( ∏_{n′=1}^{n} ρ(n′)) (\begin{matrix}z^{−1/2}u_{±,t\underline{p}} \\ z^{1/2}v_{±,t\underline{p}} \end{matrix}) + Dz^{n/2}(∂_{t\underline{p}} ∏_{n′=1}^{n} ρ(n′)) (\begin{matrix}z^{−1/2}u_{±} \\ z^{1/2}v_{±} \end{matrix}) .

Next, one observes that

(∂_{t\underline{p}} ∏_{n′=1}^{n} ρ(n′))( ∏_{n′=1}^{n} ρ(n′))^{−1} = ∂_{t\underline{p}} ln ( ∏_{n′=1}^{n} ρ(n′)) = \frac{1}{2}∂_{t\underline{p}} ln ( ∏_{n′=1}^{n} ρ(n′)^{2})

= \frac{1}{2}∂_{t\underline{p}} ln ( ∏_{n′=1}^{n} γ(n′)) = \frac{1}{2} ∑_{n′=1}^{n} \frac{γ_{t\underline{p}}(n′)}{γ(n′)}.  (5.24)

Thus, (5.23) reads

D^{−1}z^{−(n−1)/2}( ∏_{n′=1}^{n} ρ(n′))^{−1} (\begin{matrix}ψ_{1,±,t\underline{p}} \\ ψ_{2,±,t\underline{p}} \end{matrix})  (5.25)

= (\begin{matrix}u_{±,t\underline{p}} \\ zv_{±,t\underline{p}} \end{matrix}) + \frac{1}{2}( ∑_{n′=1}^{n} \frac{γ_{t\underline{p}}(n′)}{γ(n′)}) (\begin{matrix}u_{±} \\ zv_{±} \end{matrix}) + \frac{D_{t\underline{p}}}{D} (\begin{matrix}u_{±} \\ zv_{±} \end{matrix}) .  (5.26)

Combining (5.22) and (5.26) one finds that (5.21) is equivalent to

(\begin{matrix}u_{±,t\underline{p}} \\ v_{±,t\underline{p}} \end{matrix})+ \frac{1}{2}( ∑_{n′=1}^{n} \frac{γ_{t\underline{p}}(n′)}{γ(n′)}) (\begin{matrix}u_{±} \\ v_{±} \end{matrix})+ \frac{D_{t\underline{p}}}{D} (\begin{matrix}u_{±} \\ v_{±} \end{matrix}) = i (\begin{matrix}G\underline{p}u_{±} − zF\underline{p}v_{±} \\ z^{−1}H\underline{p}u_{±} − K\underline{p}v_{±} \end{matrix}) . (5.27)

Using (2.53), (2.37), and (5.16) we find

∑_{n′=1}^{n} \frac{γ_{t\underline{p}}(n′)}{γ(n′)} = i ∑_{n′=1}^{n}(I − S^{−})(g_{p_{+},+} − g_{p_{−},−})
= i((g_{p_{+},+}(n) − g_{p_{−},−}(n)) − (g_{p_{+},+}(0) − g_{p_{−},−}(0)))
= i(G\underline{p} − K\underline{p}) − 2\frac{D_{t\underline{p}}}{D} .  (5.28)

From (3.29), (3.36), (5.4), and (5.6) one obtains (we recall that n is assumed to be odd)

(\begin{matrix}u_{±,t\underline{p}} \\ v_{±,t\underline{p}} \end{matrix}) = i (\begin{matrix}−zF\underline{p}v_{±} + \frac{1}{2}(G\underline{p} + K\underline{p})u_{±} \\ −z^{−1}H\underline{p}u_{±} + \frac{1}{2}(G\underline{p} + K\underline{p})v_{±} \end{matrix}) ,  (5.29)

using (4.18).

Inserting (5.28) into (5.27), we see that it reduces to (5.29), thereby proving (5.21) in the case when n is odd. The case with n even follows from analogous computations.

Linear independence of Ψ_{−}(z, · , t\underline{p}) and Ψ_{+}(z, · , t\underline{p}) follows from

(\begin{matrix}ψ_{1,−}(z, n, t\underline{p}) & ψ_{1,+}(z, n, t\underline{p}) \\ ψ_{2,−}(z, n, t\underline{p}) & ψ_{2,+}(z, n, t\underline{p}) \end{matrix}) = D(t\underline{p})z^{n/2}( ∏_{n′=1}^{n} ρ(n′, t\underline{p}))A(z, n)^{−1}

× (\begin{matrix}u_{−}(z, n, t\underline{p}) & u_{+}(z, n, t\underline{p}) \\ v_{−}(z, n, t\underline{p}) & v_{+}(z, n, t\underline{p}) \end{matrix}) ,  (5.30)

the fact that ρ(n, t\underline{p}) ≠ 0, det(A(z, n)) = (−1)^{n+1}, and from

det ( (\begin{matrix}u_{−}(z, n, t\underline{p}) & u_{+}(z, n, t\underline{p}) \\ v_{−}(z, n, t\underline{p}) & v_{+}(z, n, t\underline{p}) \end{matrix}) ) ≠ 0, (n, t\underline{p}) ∈ ℤ × ℝ,  (5.31)

since by hypothesis z ∈ ℂ \ spec(L̆(t\underline{p})). □

In the following we will always refer to the solutions Ψ_{±} introduced in (5.14)– (5.16).

The next result recalls the existence of a propagator W\underline{p} associated with P\underline{p}. (Below we denote by B(H) the Banach space of all bounded linear operators defined on the Hilbert space H.)

Theorem 5.4. Assume Hypothesis 5.1 and suppose α, β satisfy AL\underline{p}(α, β) = 0 for some \underline{p} ∈ ℕ_{0}^{2}. Then there is a propagator W\underline{p}(s, t) ∈ B(ℓ^{2}(ℤ)), (s, t) ∈ ℝ^{2}, satisfying

(i) W\underline{p}(t, t) = I, t ∈ ℝ,  (5.32)
(ii) W\underline{p}(r, s)W\underline{p}(s, t) = W\underline{p}(r, t), (r, s, t) ∈ ℝ^{3},  (5.33)
(iii) W\underline{p}(s, t) is jointly strongly continuous in (s, t) ∈ ℝ^{2},  (5.34)

such that for fixed t_{0} ∈ ℝ, f_{0} ∈ ℓ^{2}(ℤ),

f(t) = W\underline{p}(t, t_{0})f_{0}, t ∈ ℝ,  (5.35)

satisfies

\frac{d}{dt}f(t) = P̆\underline{p}(t)f(t), f(t_{0}) = f_{0}.  (5.36)

Moreover, L̆(t) is similar to L̆(s) for all (s, t) ∈ ℝ^{2},

L̆(s) = W\underline{p}(s, t)L̆(t)W\underline{p}(s, t)^{−1}, (s, t) ∈ ℝ^{2}.  (5.37)

This extends to appropriate functions of L̆(t) and so, in particular, to its resolvent (L̆(t) − zI)^{−1}, z ∈ ℂ \ σ(L̆(t)), and hence also yields

σ(L̆(s)) = σ(L̆(t)), (s, t) ∈ ℝ^{2}.  (5.38)

Consequently, the spectrum of L̆(t) is independent of t ∈ ℝ.

Proof. (5.32)–(5.36) are standard results which follow, for instance, from Theorem X.69 of [40] under even weaker hypotheses on α, β. In particular, the propagator W\underline{p} admits the norm convergent Dyson series

W\underline{p}(s, t) = I + ∑_{k∈ℕ} ∫_{s}^{t} dt_{1} ∫_{s}^{t_{1}} dt_{2} ⋯ ∫_{s}^{t_{k−1}} dt_{k} P̆\underline{p}(t_{1})P̆\underline{p}(t_{2}) ⋯ P̆\underline{p}(t_{k}), (5.39)
(s, t) ∈ ℝ^{2}.

Fixing s ∈ ℝ and introducing the operator-valued function

K̆(t) = W\underline{p}(s, t)L̆(t)W\underline{p}(s, t)^{−1}, t ∈ ℝ,  (5.40)

one computes

K̆′(t)f = W\underline{p}(s, t)(L̆′(t) − [P̆\underline{p}(t), L̆(t)])W\underline{p}(s, t)^{−1}f = 0, t ∈ ℝ, f ∈ ℓ^{2}(ℤ),

(5.41)

using the Lax commutator equation (3.38). Thus, K̆ is independent of t ∈ ℝ and hence taking t = s in (5.40) then yields K̆ = L̆(s) and thus proves (5.37). □

Next we briefly recall the Ablowitz–Ladik initial value problem in a setting convenient for our purpose.

Theorem 5.5. Let t_{0,\underline{p}} ∈ ℝ and suppose α^{(0)}, β^{(0)} ∈ ℓ^{q}(ℤ) for some q ∈ [1, ∞) ∪ \{∞\}. Then the \underline{p}th Ablowitz–Ladik initial value problem

AL\underline{p}(α, β) = 0, (α, β)|_{t\underline{p}=t_{0,\underline{p}}} = (α^{(0)}, β^{(0)})  (5.42)

for some \underline{p} ∈ ℕ_{0}^{2}, has a unique, local, and smooth solution in time, that is, there exists a T_{0} > 0 such that

α( ·), β( ·) ∈ C^{∞}((t_{0,\underline{p}} − T_{0}, t_{0,\underline{p}} + T_{0}), ℓ^{q}(ℤ)).  (5.43)

Proof. This follows from standard results in [8, Sect. 4.1]. More precisely, local existence and uniqueness as well as smoothness of the solution of the initial value problem (5.42) (cf. (2.51)) follows from [8, Theorem 4.1.5] since f_{p_{±}−1,±}, g_{p_{±},±}, and h_{p_{±}−1,±} depend polynomially on α, β and certain of their shifts, and the fact that the Ablowitz–Ladik flows are autonomous. □

For an analogous result in connection with the Toda hierarchy we refer to [24] and [46, Sect. 12.2].

Remark 5.6. In the special defocusing case, where β = \overline{α} and hence L̆(t), t ∈ ℝ, is unitary, one obtains

sup_{(n,t\underline{p})∈ℕ×(t_{0,\underline{p}}−T_{0},t_{0,\underline{p}}+T_{0})} |α(n, t\underline{p})| ≤ 1  (5.44)

using γ = 1−|α|^{2} and γ_{t\underline{p}} = iγ((g_{p_{+},+}−g_{p_{+},+}^{−})−(g_{p_{−},−}−g_{p_{−},−}^{−})) in (2.53). A further application of [8, Proposition 4.1.22] then yields a unique, global, and smooth solution of the \underline{p}th AL initial value problem (5.42). Moreover, the same argument shows that if α satisfies Hypothesis 5.1 and the \underline{p}th AL equation AL\underline{p}(α, \overline{α}) = 0, then α is actually smooth with respect to t\underline{p} ∈ ℝ, that is,

α(n, ·) ∈ C^{∞}(ℝ), n ∈ ℤ.  (5.45)

Equation (5.18), that is, Ψ_{±,t\underline{p}} = V + ; \underline{p}Ψ_{±}, implies that

∂_{t\underline{p}} ln (\frac{ψ_{1,±}^{+}}{ψ_{1,±}} ) = (S^{+} − I)∂_{t\underline{p}} ln(ψ_{1,±})

= (S^{+} − I)\frac{∂_{t\underline{p}}ψ_{1,±}}{ψ_{1,±}}

= i(S^{+} − I)(G\underline{p} − F\underline{p}ϕ_{±}).  (5.46)

On the other hand, equation (5.17), that is, UΨ_{±}^{−} = Ψ_{±}, yields

∂_{t\underline{p}} ln (\frac{ψ_{1,±}^{+}}{ψ_{1,±}} ) = ∂_{t\underline{p}} ln(z + α^{+}ϕ_{±}),  (5.47)

and thus one concludes that

∂_{t\underline{p}} ln(z + α^{+}ϕ_{±}) = i(S^{+} − I)(G\underline{p} − F\underline{p}ϕ_{±}).  (5.48)

Below we will refer to (5.48±) according to the upper or lower sign in (5.48). Expanding (5.48±) in powers of z and 1/z then yields the following conserved densities:

Theorem 5.7. Assume Hypothesis 5.1 and suppose α, β satisfy AL\underline{p}(α, β) = 0 for some \underline{p} ∈ ℕ_{0}^{2}. Then the following infinite sequences of local conservation laws hold:

Expansion of (5.48+) at 1/z = 0:

∂_{t\underline{p}}ρ_{j,+}^{∞} = i(S^{+} − I)(g_{p_{−}−j,−} − ∑_{ℓ=0}^{j−1} f_{p_{−}−j+ℓ,−}ϕ_{ℓ,+}^{∞} − ∑_{ℓ=0}^{p_{+}−1} f_{p_{+}−1−ℓ,+}ϕ_{j+ℓ,+}^{∞}),
j = 1, …, p_{−}, (5.49)
∂_{t\underline{p}}ρ_{j,+}^{∞} = −i(S^{+} − I)( ∑_{ℓ=1}^{p_{−}} f_{p_{−}−ℓ,−}ϕ_{j−ℓ,+}^{∞} + ∑_{ℓ=0}^{p_{+}−1} f_{p_{+}−1−ℓ,+}ϕ_{j+ℓ,+}^{∞}),
j ≥ p_{−} + 1,  (5.50)

where ρ_{j,+}^{∞} and ϕ_{j,+}^{∞} are given by (4.48) and (4.38).

Expansion of (5.48−) at 1/z = 0:

∂_{t\underline{p}}ρ_{j,−}^{∞} = i(S^{+} − I)(g_{p_{−}−j,−} − ∑_{ℓ=−1}^{j−1} f_{p_{−}+ℓ−j,−}ϕ_{ℓ,−}^{∞} − ∑_{ℓ=0}^{p_{+}−1} f_{p_{+}−1−ℓ,+}ϕ_{j+ℓ,−}^{∞}),
j = 1, …, p_{−}, (5.51)
∂_{t\underline{p}}ρ_{j,−}^{∞} = −i(S^{+} − I)( ∑_{ℓ=1}^{p_{−}} f_{p_{−}−ℓ,−}ϕ_{j−ℓ,−}^{∞} + ∑_{ℓ=0}^{p_{+}−1} f_{p_{+}−1−ℓ,+}ϕ_{j+ℓ,−}^{∞}),
j ≥ p_{−} + 1,  (5.52)

where ρ_{j,−}^{∞} and ϕ_{j,−}^{∞} are given by (4.50) and (4.39).

Expansion of (5.48+) at z = 0:

∂_{t\underline{p}}ρ_{j,+}^{0} = i(S^{+} − I)(g_{p_{+}−j,+} − ∑_{ℓ=1}^{p_{−}} ϕ_{j+ℓ,+}^{0}f_{p_{−}−ℓ,−} − ∑_{ℓ=0}^{j} ϕ_{ℓ,+}^{0}f_{p_{+}−1−j+ℓ,+}),
j = 1, …, p_{+} − 1, (5.53)
∂_{t\underline{p}}ρ_{p_{+},+}^{0} = i(S^{+} − I)(g_{0,+} − ∑_{ℓ=1}^{p_{−}} ϕ_{j+ℓ,+}^{0}f_{p_{−}−ℓ,−} − ∑_{ℓ=0}^{p_{+}−1} ϕ_{j+ℓ−p_{+}+1,+}^{0}f_{ℓ,+}), (5.54)

∂_{t\underline{p}}ρ_{j,+}^{0} = −i(S^{+} − I)( ∑_{ℓ=1}^{p_{−}} ϕ_{j+ℓ,+}^{0}f_{p_{−}−ℓ,−} + ∑_{ℓ=0}^{p_{+}−1} ϕ_{j+ℓ−p_{+}+1,+}^{0}f_{ℓ,+}),
j ≥ p_{+} + 1,  (5.55)

where ρ_{j,+}^{0} and ϕ_{j,+}^{0} are given by (4.52) and (4.40).

Expansion of (5.48−) at z = 0:

∂_{t\underline{p}}ρ_{j,−}^{0} = i(S^{+} − I)(g_{p_{+}−j,+} − ∑_{ℓ=1}^{j} ϕ_{ℓ,−}^{0}f_{p_{+}−j+ℓ−1,+} − ∑_{ℓ=1}^{p_{−}} ϕ_{j+ℓ,−}^{0}f_{p_{−}−ℓ,−}),
j = 1, …, p_{+}, (5.56)
∂_{t\underline{p}}ρ_{j,−}^{0} = −i(S^{+} − I)( ∑_{ℓ=j+1−p_{+}}^{j} ϕ_{ℓ,−}^{0}f_{p_{+}−j+ℓ−1,+} + ∑_{ℓ=1}^{p_{−}} ϕ_{j+ℓ,−}^{0}f_{p_{−}−ℓ,−}),
j ≥ p_{+} + 1, (5.57)

where ρ_{j,−}^{0} and ϕ_{j,−}^{0} are given by (4.54) and (4.41).

Proof. The proof consists of expanding (5.48±) in powers of z and 1/z and applying (4.47)–(4.54).

Expansion of (5.48+) at 1/z = 0: For the right-hand side of (5.48+) one finds

G\underline{p} − F\underline{p}ϕ_{+} = ∑_{ℓ=1}^{p_{−}} g_{p_{−}−ℓ,−}z^{−ℓ} + ∑_{ℓ=0}^{p_{+}} g_{p_{+}−ℓ,+}z^{ℓ}

− ( ∑_{ℓ=1}^{p_{−}} f_{p_{−}−ℓ,−}z^{−ℓ} + ∑_{ℓ=0}^{p_{+}−1} f_{p_{+}−1−ℓ,+}z^{ℓ}) ∑_{j=0}^{∞} ϕ_{j,+}^{∞}z^{−j}

= g_{0,+}z^{p_{+}} + ∑_{j=0}^{p_{+}−1} (g_{p_{+}−j,+} − ∑_{ℓ=0}^{p_{+}−j−1} f_{p_{+}−j−1−ℓ,+}ϕ_{ℓ,+}^{∞})z^{j}

+ ∑_{j=1}^{p_{−}} (g_{p_{−}−j,−} − ∑_{ℓ=0}^{j−1} f_{p_{−}−j+ℓ,−}ϕ_{ℓ,+}^{∞} − ∑_{ℓ=0}^{p_{+}−1} f_{p_{+}−1−ℓ,+}ϕ_{j+ℓ,+}^{∞})z^{−j}

− ∑_{j=p_{−}+1}^{∞} ( ∑_{ℓ=1}^{p_{−}} f_{p_{−}−ℓ,−}ϕ_{j−ℓ,+}^{∞} + ∑_{ℓ=0}^{p_{+}−1} f_{p_{+}−1−ℓ,+}ϕ_{j+ℓ,+}^{∞})z^{−j}. (5.58)

Here we used that that all positive powers vanish because of (5.48). This yields the following additional formulas:

Conservation laws derived from ϕ_{+} at 1/z = 0:

(S^{+} − 1)(g_{p_{+}−j,+} − ∑_{ℓ=0}^{p_{+}−j−1} f_{p_{+}−j−1−ℓ,+}ϕ_{ℓ,+}^{∞}) = 0, j = 0, …, p_{+} − 1,

(S^{+} − 1)g_{0,+} = 0.  (5.59)

Expansion of (5.48−) at 1/z = 0: The right-hand side of (5.48−) yields

G\underline{p} − F\underline{p}ϕ_{−} = ∑_{ℓ=1}^{p_{−}} g_{p_{−}−ℓ,−}z^{−ℓ} + ∑_{ℓ=0}^{p_{+}} g_{p_{+}−ℓ,+}z^{ℓ}

− ( ∑_{ℓ=1}^{p_{−}} f_{p_{−}−ℓ,−}z^{−ℓ} + ∑_{ℓ=0}^{p_{+}−1} f_{p_{+}−1−ℓ,+}z^{ℓ}) ∑_{j=−1}^{∞} ϕ_{j,−}^{∞}z^{−j}

= ∑_{j=1}^{p_{+}} (g_{p_{+}−j,+} − ∑_{ℓ=−1}^{p_{+}−j−1} f_{p_{+}−j−1−ℓ,+}ϕ_{ℓ,−}^{∞})z^{j}

+ (g_{p_{+},+} − ∑_{ℓ=0}^{p_{+}−1} f_{p_{+}−1−ℓ,+}ϕ_{ℓ,−}^{∞} − f_{p_{−}−1,−}ϕ_{−1,−}^{∞})

+ ∑_{j=1}^{p_{−}} (g_{p_{−}−j,−} − ∑_{ℓ=−1}^{j−1} f_{p_{−}+ℓ−j,−}ϕ_{ℓ,−}^{∞} − ∑_{ℓ=0}^{p_{+}−1} f_{p_{+}−1−ℓ,+}ϕ_{j+ℓ,−}^{∞})z^{−j}

− ∑_{j=p_{−}+1}^{∞} ( ∑_{ℓ=1}^{p_{−}} f_{p_{−}−ℓ,−}ϕ_{j−ℓ,−}^{∞} + ∑_{ℓ=0}^{p_{+}−1} f_{p_{+}−1−ℓ,+}ϕ_{j+ℓ,−}^{∞})z^{−j}. (5.60)

Conservation laws derived from ϕ_{+} at 1/z = 0:

(S^{+} − I)(g_{p_{+}−j,+} − ∑_{ℓ=−1}^{p_{+}−j−1} f_{p_{+}−j−1−ℓ,+}ϕ_{ℓ,−}^{∞}) = 0, j = 1, …, p_{+}, (5.61)

i(S^{+} − I)(g_{p_{+},+} − ∑_{ℓ=0}^{p_{+}−1} f_{p_{+}−1−ℓ,+}ϕ_{ℓ,−}^{∞} − f_{p_{−}−1,−}ϕ_{−1,−}^{∞})

= ∂_{t\underline{p}} ln (\frac{α^{++}}{α^{+}} ) + ∂_{t\underline{p}} ln(γ^{+}).  (5.62)

Expansion of (5.48+) at z = 0: For the right-hand side of (5.48+) one finds

G\underline{p} − F\underline{p}ϕ_{+} = ∑_{j=1}^{p_{−}} (g_{p_{−}−j,−} − ∑_{ℓ=0}^{p_{−}−j} ϕ_{ℓ,+}f_{p_{−}−j−ℓ,−})z^{−j}  (5.63)

+ ∑_{j=0}^{p_{+}−1} (g_{p_{+}−j,+} − ∑_{ℓ=1}^{p_{−}} ϕ_{j+ℓ,+}^{0}f_{p_{−}−ℓ,−} − ∑_{ℓ=0}^{j} ϕ_{ℓ,+}^{0}f_{p_{+}−1−j+ℓ,+})z^{j}

+ ∑_{j=p_{+}}^{∞} (g_{0,+}χ_{jp_{+}} − ∑_{ℓ=1}^{p_{−}} ϕ_{j+ℓ,+}^{0}f_{p_{−}−ℓ,−} − ∑_{ℓ=0}^{p_{+}−1} ϕ_{j+ℓ−p_{+}+1,+}^{0}f_{ℓ,+})z^{j}.

Conservation laws derived from ϕ_{+} at z = 0:

(S^{+} − I)(g_{j,−} − ∑_{ℓ=0}^{j} ϕ_{ℓ,+}^{0}f_{j−ℓ,−}) = 0, j = 1, …, p_{−},  (5.64)

(S^{+} − I)(g_{p_{+},+} − ϕ_{0,+}^{0}f_{p_{+}−1,+} − ∑_{ℓ=1}^{p_{−}} ϕ_{ℓ,+}^{0}f_{p_{−}−ℓ,−}) = ∂_{t\underline{p}} ln (\frac{α}{α^{+}} ).  (5.65)

Expansion of (5.48−) at z = 0: For the right-hand side of (5.48−) one finds

G\underline{p} − F\underline{p}ϕ_{−} = g_{0,−}z^{−p_{−}} + ∑_{j=1}^{p_{−}−1} (g_{p_{−}−j,−} − ∑_{ℓ=1}^{p_{−}−j} ϕ_{ℓ,−}^{0}f_{p_{−}−j−ℓ,−})z^{−j}  (5.66)

+ g_{p_{+},+} − ∑_{ℓ=1}^{p_{−}} ϕ_{ℓ,−}^{0}f_{p_{−}−ℓ,−}

+ ∑_{j=1}^{p_{+}} (g_{p_{+}−j,+} − ∑_{ℓ=1}^{j} ϕ_{ℓ,−}^{0}f_{p_{+}−j+ℓ−1,+} − ∑_{ℓ=1}^{p_{−}} ϕ_{j+ℓ,−}^{0}f_{p_{−}−ℓ,−})z^{j}

− ∑_{j=p_{+}+1}^{∞} ( ∑_{ℓ=j+1−p_{+}}^{j} ϕ_{ℓ,−}^{0}f_{p_{+}−j+ℓ−1,+} + ∑_{ℓ=1}^{p_{−}} ϕ_{j+ℓ,−}^{0}f_{p_{−}−ℓ,−})z^{j}.

Conservation laws derived from ϕ_{−} at z = 0:

(S^{+} − I)g_{0,−} = 0,  (5.67)
(S^{+} − I)(g_{j,−} − ∑_{ℓ=1}^{j} ϕ_{ℓ,−}^{0}f_{j−ℓ,−}) = 0, j = 1, …, p_{−} − 1,  (5.68)

(S^{+} − I)(g_{p_{+},+} − ∑_{ℓ=1}^{p_{−}} ϕ_{ℓ,−}^{0}f_{p_{−}−ℓ,−}) = ∂_{t\underline{p}} ln(γ^{+}).  (5.69)

Combining these expansions with (4.47)–(4.53) finishes the proof. □

Remark 5.8. (i) There is a certain redundancy in the conservation laws (5.49)– (5.57) as can be observed from Lemma 4.4. Equations (4.56)–(4.57) imply

ρ_{j,+}^{∞} = −ρ_{j,−}^{∞} + \frac{1}{j}(S^{+} − I)(d_{j,+} − e_{j,+}), j ∈ ℕ,  (5.70)
ρ_{j,+}^{0} = −ρ_{j,−}^{0} + \frac{1}{j}(S^{+} − I)(d_{j,−} − e_{j,−}), j ∈ ℕ.  (5.71)

Thus one can, for instance, transfer (5.49)–(5.50) into (5.51)–(5.52). (ii) In addition to the conservation laws listed in Theorem 5.7, we recover the familiar conservation law (cf. (2.53))

∂_{t\underline{p}} ln(γ) = i(I − S^{−})(g_{p_{+},+} − g_{p_{−},−}), \underline{p} ∈ ℕ_{0}^{2}.  (5.72)

(iii) Another consequence of Theorem 5.7 and Lemma 4.4 is that for α, β satisfying Hypothesis 5.1 and α, β ∈ C^{1}(ℝ, ℓ^{2}(ℤ)), one has

\frac{d}{dt\underline{p}} ∑_{n∈ℤ} ln(γ(n, t\underline{p})) = 0, \frac{d}{dt\underline{p}} ∑_{n∈ℤ} ĝ_{j,±}(n, t\underline{p}) = 0, j ∈ ℕ, \underline{p} ∈ ℕ_{0}^{2}.  (5.73)

Remark 5.9. The infinite sequence of conservation laws has been studied in the literature; we refer to [4], [7, Ch. 3], [14], [50], and [52]. In particular, Zhang and Chen [50] study local conservation laws for the full 4 × 4 Ablowitz–Ladik system in a similar way to the one employed here. However, they only expand their equation around a point that corresponds to 1/z = 0. The systematic derivation of infinite sequences of conserved densities and currents (cf. the corresponding discussion in the introduction) as presented in Theorem 5.7 appears to be new.

The two local conservation laws coming from expansions around z = 0 are essentially the same since the two conserved densities, ρ_{j,+}^{0} and ρ_{j,+}^{0}, differ by a first-order difference expression (cf. Remark 5.8). A similar argument applies to the expansions around 1/z = 0. That there are two independent sequences of conservation laws is also clear from (5.73) which yields that ∑_{n∈ℤ} ĝ_{j,±}(n, t\underline{p}) are time-independent. One observes that the quantities ĝ_{j,+}, j ∈ ℕ, are related to the expansions around 1/z = 0, that is, to ρ_{j,±}^{∞}, while ĝ_{j,−}, j ∈ ℕ, are related to ρ_{j,±}^{0} (cf. Lemma 4.4). In addition to the two infinite sequences of polynomial conservation laws, there is a logarithmic conservation law (cf. (5.72) and (5.73)).

The first conservation laws explicitly read as follows: p_{+} = p_{−} = 1:

∂_{t_{(1,1)}}ρ_{j,±}^{∞} = −i(S^{+} − I)(f_{0,−}ϕ_{j−1,±}^{∞} + f_{0,+}ϕ_{j,±}^{∞}), j ≥ 1,  (5.74)
∂_{t_{(1,1)}}ρ_{j,±}^{0} = −i(S^{+} − I)(f_{0,−}ϕ_{j+1,±}^{0} + f_{0,+}ϕ_{j,±}^{0}), j ≥ 1.  (5.75)

For j = 1 this yields using (4.55)

∂_{t_{(1,1)}}ρ_{1,+}^{∞} = ∂_{t_{(1,1)}}α^{+}β = i(S^{+} − I)(−c_{0,−}αβ + c_{0,+}α^{+}β^{−}γ),
∂_{t_{(1,1)}}ρ_{1,−}^{∞} = ∂_{t_{(1,1)}}( − α^{+++}β^{++} + (S^{+} − I)\frac{α^{++}}{α^{+}} )

= i(S^{+} − I)(c_{0,+} \frac{α^{+++}}{α^{+}} γ^{+}γ^{++} − c_{0,−} \frac{αα^{++}}{(α^{+})^{2}} γ^{+} − c_{0,+}(\frac{αα^{++}}{α^{+}} )^{2}γ^{+}),

∂_{t_{(1,1)}}ρ_{1,+}^{0} = ∂_{t_{(1,1)}}(α^{−}β + (S^{+} − I)\frac{α^{−}}{α} )  (5.76)

= i(S^{+} − I)(c_{0,−} \frac{α^{−−}}{α} γ^{−}γ − c_{0,+} \frac{α^{−}α^{+}}{α^{2}} γ − c_{0,−}(\frac{α^{−}}{α} )^{2}γ),

∂_{t_{(1,1)}}ρ_{1,−}^{0} = ∂_{t_{(1,1)}}α^{+}β^{++} = i(S^{+} − I)(c_{0,+}α^{+}β^{+} − c_{0,−}αβ^{++}γ^{+}).

This shows in particular that we obtain two sets of conservation laws (one from expanding near ∞ and the other from expanding near 0), where the first few equations of each set explicitly read (p_{+} = p_{−} = 1):

\begin{matrix}j = 1 : ∂_{t_{(1,1)}}α^{+}β = i(S^{+} − I)(−c_{0,−}αβ + c_{0,+}α^{+}β^{−}γ), \\ ∂_{t_{(1,1)}}αβ^{+} = i(S^{+} − I)(c_{0,+}αβ − c_{0,−}α^{−}β^{+}γ), \end{matrix}  (5.77)
j = 2 : ∂_{t_{(1,1)}}( − \frac{1}{2}(α^{+}β)^{2} + γα^{+}β^{−})
= i(S^{+} − I)γ( − c_{0,−}αβ^{−} − c_{0,+}αα^{+}(β^{−})^{2} + c_{0,+}γ^{−}α^{+}β^{−−}),

∂_{t_{(1,1)}}(\frac{1}{2}(αβ^{+})^{2} − γ^{+}αβ^{++})  (5.78)
= i(S^{+} − I)γ( − c_{0,+}αβ^{+} − c_{0,−}α^{−}α(β^{+})^{2} + c_{0,−}γ^{+}α^{−}β^{++}).

Using Lemma 4.4, one observes that one can replace ρ_{j,±}^{∞,0} in Theorem 5.7 by ĝ_{j,±} by suitably adjusting the right-hand sides in (5.49)–(5.57).

6. Hamiltonian formalism, variational derivatives

We start this section by a short review of variational derivatives for discrete systems. Consider the functional

\begin{matrix}G: ℓ^{1}(ℤ)^{κ} → ℂ, \\ G(u) = ∑_{n∈ℤ} G(u(n), u^{(+1)}(n), u^{(−1)}(n), …, u^{(k)}(n), u^{(−k)}(n)) \end{matrix}  (6.1)

for some κ ∈ ℕ and k ∈ ℕ_{0}, where G: ℤ × ℂ^{2rκ} → ℂ is C^{1} with respect to the 2rκ complex-valued entries and where

u^{(s)} = S^{(s)}u, S^{(s)} = \begin{cases}(S^{+})^{s}u & if s ≥ 0, \\ (S^{−})^{−s}u & if s < 0, \end{cases} u ∈ ℓ^{∞}(ℤ)^{κ}.  (6.2)

For brevity we write

G(u(n)) = G(u(n), u^{(+1)}(n), u^{(−1)}(n), …, u^{(k)}(n), u^{(−k)}(n)).  (6.3)

The functional G is Frechet-differentiable and one computes for any v ∈ ℓ^{1}(ℤ)^{κ} for the differential dG

(dG)_{u}(v) = \frac{d}{dϵ}G(u + ϵv)|_{ϵ=0}

= ∑_{n∈ℤ} (\frac{∂G(u(n))}{∂u}v(n) + \frac{∂G(u(n))}{∂u^{(+1)}}v^{(+1)}(n) + \frac{∂G(u(n))}{∂u^{(−1)}}v^{(−1)}(n)

+ ⋯ + \frac{∂G(u(n))}{∂u^{(k)}}v^{(k)}(n) + \frac{∂G(u(n))}{∂u^{(−k)}}v^{(−k)}(n))

= ∑_{n∈ℤ} (\frac{∂G(u(n))}{∂u} + S^{(−1)} \frac{∂G(u(n))}{∂u^{(+1)}} + S^{(+1)} \frac{∂G(u(n))}{∂u^{(−1)}}

+ ⋯ + S^{(−k)} \frac{∂G(u(n))}{∂u^{(k)}} + S^{(k)} \frac{∂G(u(n))}{∂u^{(−k)}})v(n)

= ∑_{n∈ℤ} \frac{δG}{δu}(n)v(n),  (6.4)

where we introduce the gradient and the variational derivative of G by

(∇G)_{u} = \frac{δG}{δu}  (6.5)

= \frac{∂G}{∂u} + S^{(−1)} \frac{∂G}{∂u^{(+1)}} + S^{(+1)} \frac{∂G}{∂u^{(−1)}} + ⋯ + S^{(−k)} \frac{∂G}{∂u^{(k)}} + S^{(k)} \frac{∂G}{∂u^{(−k)}} ,

assuming

\{G(u(n))\}_{n∈ℤ}, \{\frac{∂G(u(n))}{∂u^{(±j)}}\}_{n∈ℤ} ∈ ℓ^{1}(ℤ), j = 1, …, k.  (6.6)

To establish the connection with the Ablowitz–Ladik hierarchy we make the following assumption for the remainder of this section.

Hypothesis 6.1. Suppose

α, β ∈ ℓ^{1}(ℤ), α(n)β(n) ∉ {0, 1}, n ∈ ℤ.  (6.7)

Next, let G be a functional of the type

G: ℓ^{∞}(ℤ)^{2} → ℂ,

G(α, β) = ∑_{n∈ℤ} G(α(n), β(n), …, α(n + k), β(n + k), α(n − k), β(n − k))

= ∑_{n∈ℤ} G(α(n), β(n)),  (6.8)

where G(α, β) is polynomial in α, β and some of their shifts. The gradient ∇G and symplectic gradient ∇_{s}G of G are then defined by

(∇G)_{α,β} = (\begin{matrix}(∇G)_{α} \\ (∇G)_{β} \end{matrix}) = (\frac{δG}{δα} ; \frac{δG}{δβ})  (6.9)

and

(∇_{s}G)_{α,β} = D(∇G)_{α,β} = D (\begin{matrix}(∇G)_{α} \\ (∇G)_{β} \end{matrix}) ,  (6.10)

respectively. Here D is defined by

D = γ (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) , γ = 1 − αβ.  (6.11)

In addition, we introduce the bilinear form

Ω: ℓ^{1}(ℤ)^{2} × ℓ^{1}(ℤ)^{2} → ℂ,
Ω(u, v) = ∑_{n∈ℤ}(D^{−1}u)(n) · v(n).  (6.12)

One then concludes that

\begin{aligned}Ω(Du, v) & = ∑_{n∈ℤ} u(n) · v(n) = ∑_{n∈ℤ} (u_{1}(n)v_{1}(n) + u_{2}(n)v_{2}(n)) \\ & = ⟨u, v⟩_{ℓ^{2}(ℤ)^{2}}, u, v ∈ ℓ^{1}(ℤ)^{2}, \end{aligned}  (6.13)

where ⟨ · , · ⟩_{ℓ^{2}(ℤ)^{2}} denotes the “real” inner product in ℓ^{2}(ℤ)^{2}, that is,

\begin{matrix}⟨ · , · ⟩_{ℓ^{2}(ℤ)^{2}} : ℓ^{2}(ℤ)^{2} × ℓ^{2}(ℤ)^{2} → ℂ, \\ ⟨u, v⟩_{ℓ^{2}(ℤ)^{2}} = ∑_{n∈ℤ} u(n) · v(n) = ∑_{n∈ℤ} (u_{1}(n)v_{1}(n) + u_{2}(n)v_{2}(n)). \end{matrix}  (6.14)

In addition, one obtains

(dG)_{α,β}(v) = ⟨(∇G)_{α,β}, v⟩_{ℓ^{2}(ℤ)^{2}} = Ω(D(∇G)_{α,β}, v) = Ω((∇_{s}G)_{α,β}, v). (6.15)

Given two functionals G_{1}, G_{2} we define their Poisson bracket by

{G_{1}, G_{2}} = dG_{1}(∇_{s}G_{2}) = Ω(∇_{s}G_{1}, ∇_{s}G_{2})
= Ω(D∇G_{1}, D∇G_{2}) = ⟨∇G_{1}, D∇G_{2}⟩_{ℓ^{2}(ℤ)^{2}}
= ∑_{n∈ℤ} (\begin{matrix}\frac{δG_{1}}{δα} (n) \\ \frac{δG_{1}}{δβ} (n) \end{matrix}) · D (\begin{matrix}\frac{δG_{2}}{δα} (n) \\ \frac{δG_{2}}{δβ} (n) \end{matrix}) .  (6.16)

Since Ω( · , ·) is a weakly non-degenerate closed 2-form, both the Jacobi identity

{{G_{1}, G_{2}}, G_{3}} + {{G_{2}, G_{3}}, G_{1}} + {{G_{3}, G_{1}}, G_{2}} = 0,  (6.17)

as well as the Leibniz rule

{G_{1}, G_{2}G_{3}} = {G_{1}, G_{2}}G_{3} + G_{2}{G_{1}, G_{3}},  (6.18)

hold as discussed in [32, Theorem 48.8].

If G is a smooth functional and (α, β) develops according to a Hamiltonian flow with Hamiltonian H, that is,

(\begin{matrix}α \\ β \end{matrix})_{t} = (∇_{s}H)_{α,β} = D(∇H)_{α,β} = D (\frac{δH}{δα} ; \frac{δH}{δβ} ) ,  (6.19)

then

\frac{dG}{dt} = \frac{d}{dt} ∑_{n∈ℤ} G(α(n), β(n))

= ∑_{n∈ℤ} (\begin{matrix}\frac{δG}{δα}(n) \\ \frac{δG}{δβ}(n) \end{matrix}) · (\begin{matrix}α(n) \\ β(n) \end{matrix})_{t} = ∑_{n∈ℤ} (\begin{matrix}\frac{δG}{δα}(n) \\ \frac{δG}{δβ}(n) \end{matrix}) · D (\begin{matrix}\frac{δH}{δα} (n) \\ \frac{δH}{δβ} (n) \end{matrix})

= {G, H}.  (6.20)

Here, and in the remainder of this section, time-dependent equations such as (6.20) are viewed locally in time, that is, assumed to hold on some open t-interval 𝕀 ⊆ ℝ.

If a functional G is in involution with the Hamiltonian H, that is,

{G, H} = 0,  (6.21)

then it is conserved in the sense that

\frac{dG}{dt} = 0.  (6.22)

Next, we turn to the specifics of the AL hierarchy. We define

Ĝ_{ℓ,±} = ∑_{n∈ℤ} ĝ_{ℓ,±}(n).  (6.23)

Lemma 6.2. Assume Hypothesis 6.1 and v ∈ ℓ^{1}(ℤ). Then,

(dĜ_{ℓ,±})_{β}(v) = ∑_{n∈ℤ} \frac{δĝ_{ℓ,±}(n)}{δβ}v(n) = ±ℓ ∑_{n∈ℤ}(δ_{n}, L^{±ℓ−1}M_{β}(v)δ_{n}), ℓ ∈ ℕ, (6.24)

(dĜ_{ℓ,±})_{α}(v) = ∑_{n∈ℤ} \frac{δĝ_{ℓ,±}(n)}{δα}v(n) = ±ℓ ∑_{n∈ℤ}(δ_{n}, L^{±ℓ−1}M_{α}(v)δ_{n}), ℓ ∈ ℕ, (6.25)

where

M_{β}(v) = −vα^{+} + ((v^{−}ρ − β^{−} \frac{vα}{2ρ})δ_{even} + α^{+} \frac{vα}{2ρ}δ_{odd})S^{−}
+ ((vρ^{+} − β \frac{v^{+}α^{+}}{2ρ^{+}} )δ_{even} + α^{++} \frac{v^{+}α^{+}}{2ρ^{+}} δ_{odd})S^{+}  (6.26)

− (ρ\frac{v^{−}α^{−}}{2ρ^{−}} + ρ^{−} \frac{vα}{2ρ})δ_{even}S^{−−} − (ρ^{+} \frac{v^{++}α^{++}}{2ρ^{++}} + ρ^{++} \frac{v^{+}α^{+}}{2ρ^{+}} )δ_{odd}S^{++},

M_{α}(v) = −v^{+}β − ((v^{+}ρ − α^{+} \frac{vβ}{2ρ} )δ_{odd} + β^{−} \frac{vβ}{2ρ} δ_{even})S^{−}

− ((v^{++}ρ^{+} − α^{++} \frac{v^{+}β^{+}}{2ρ^{+}} )δ_{odd} − β \frac{v^{+}β^{+}}{2ρ^{+}} δ_{even})S^{+}  (6.27)

− (ρ\frac{v^{−}β^{−}}{2ρ^{−}} + ρ^{−} \frac{vβ}{2ρ} )δ_{even}S^{−−} − (ρ^{+} \frac{v^{++}β^{++}}{2ρ^{++}} + ρ^{++} \frac{v^{+}β^{+}}{2ρ^{+}} )δ_{odd}S^{++}.

Proof. We first consider the derivative with respect to β. By a slight abuse of notation we write L = L(β). Using (6.23) and (3.21) one finds

(dĜ_{ℓ,±})_{β}v = \frac{d}{dϵ}G(β + ϵv)|_{ϵ=0} = ∑_{n∈ℤ} \frac{d}{dϵ}ĝ_{ℓ,±}(β + ϵv)(n)|_{ϵ=0}

= ∑_{n∈ℤ}(δ_{n}, \frac{d}{dϵ}L(β + ϵv)^{±ℓ}δ_{n})|_{ϵ=0}.  (6.28)

Next, one considers

\frac{d}{dϵ}L(β + ϵv)^{ℓ}|_{ϵ=0} = lim_{ϵ→0} \frac{1}{ϵ}(L(β + ϵv)^{ℓ} − L(β)^{ℓ})

= lim_{ϵ→0} \frac{1}{ϵ}((L(β + ϵv) − L(β))L(β)^{ℓ−1}

+ L(β)(L(β + ϵv) − L(β))L(β)^{ℓ−2}
+ ⋯ + L(β)^{ℓ−1}(L(β + ϵv) − L(β)))
= M_{β}L(β)^{ℓ−1} + L(β)M_{β}L(β)^{ℓ−2} + ⋯ + L(β)^{ℓ−1}M_{β}, (6.29)

where

M_{β} = lim_{ϵ→0} \frac{1}{ϵ}(L(β + ϵv) − L(β))

= ( − v(n)α(n + 1)δ_{m,n} + ((v(n − 1)ρ(n) − β(n − 1)\frac{v(n)α(n)}{2ρ(n)})δ_{odd}(n)

+ α(n + 1)\frac{v(n)α(n)}{2ρ(n)}δ_{even}(n))δ_{m,n−1}

+ ((v(n)ρ(n + 1) − β(n)\frac{v(n + 1)α(n + 1)}{2ρ(n + 1)})δ_{odd}(n)  (6.30)

+ α(n + 2)\frac{v(n + 1)α(n + 1)}{2ρ(n + 1)}δ_{even}(n))δ_{m,n+1}

− (ρ(n + 1)\frac{v(n + 2)α(n + 2)}{2ρ(n + 2)} + ρ(n + 2)\frac{v(n + 1)α(n + 1)}{2ρ(n + 1)})δ_{even}(n)δ_{m,n+2}

− (ρ(n)\frac{v(n − 1)α(n − 1)}{2ρ(n − 1)} + ρ(n − 1)\frac{v(n)α(n)}{2ρ(n)})δ_{odd}(n)δ_{m,n−2})_{m,n∈ℤ}.

Similarly one obtains

\frac{d}{dϵ}L(β + ϵv)^{−ℓ}|_{ϵ=0}  (6.31)

= −(L(β)^{−1}M_{β}L(β)^{−ℓ} + L(β)^{−2}M_{β}L(β)^{−ℓ+1} + ⋯ + L(β)^{−ℓ}M_{β}L(β)^{−1}).

Inserting the expression (6.29) into (6.28) one finds

(dĜ_{ℓ,+})_{β}v = ∑_{n∈ℤ}(δ_{n}, \frac{d}{dϵ}L(β + ϵv)^{ℓ}δ_{n})|_{ϵ=0}

= ∑_{n∈ℤ}(δ_{n}, ∑_{k=0}^{ℓ−1} L^{k}M_{β}L^{ℓ−1−k}δ_{n})

= ∑_{k=0}^{ℓ−1} ∑_{n∈ℤ}(δ_{n}, L^{k}M_{β}L^{ℓ−1−k}δ_{n})

= ∑_{k=0}^{ℓ−1} ∑_{n∈ℤ}(δ_{n}, (L^{ℓ−1}M_{β} + [L^{k}M_{β}, L^{ℓ−1−k}])δ_{n})

= ∑_{k=0}^{ℓ−1} ∑_{n∈ℤ} ((δ_{n}, L^{ℓ−1}M_{β}δ_{n}) + (δ_{n}, [L^{k}M_{β}, L^{ℓ−1−k}]δ_{n}))

= ℓ ∑_{n∈ℤ}(δ_{n}, L^{ℓ−1}M_{β}δ_{n}) + ∑_{k=0}^{ℓ−1} tr ([L^{k}M_{β}, L^{ℓ−1−k}])

= ℓ ∑_{n∈ℤ}(δ_{n}, L^{ℓ−1}M_{β}δ_{n}).  (6.32)

Similarly, using (6.28) and (6.31), one concludes that

(dĜ_{ℓ,−})_{β}v = −ℓ ∑_{n∈ℤ}(δ_{n}, L^{−ℓ−1}M_{β}δ_{n}).  (6.33)

For the derivative with respect to α we set L = L(α) and replace M_{β} by

M_{α} = lim_{ϵ→0} \frac{1}{ϵ}(L(α + ϵv) − L(α))

= ( − v(n + 1)β(n)δ_{m,n} + ( − β(n − 1)\frac{v(n)β(n)}{2ρ(n)}δ_{odd}(n)

− (v(n + 1)ρ(n) − α(n + 1)\frac{v(n)β(n)}{2ρ(n)})δ_{even}(n))δ_{m,n−1}

− (β(n)\frac{v(n + 1)β(n + 1)}{2ρ(n + 1)}δ_{odd}(n) + (v(n + 2)ρ(n + 1)  (6.34)

+ α(n + 2)\frac{v(n + 1)β(n + 1)}{2ρ(n + 1)})δ_{even}(n))δ_{m,n+1}

− (ρ(n + 1)\frac{v(n + 2)β(n + 2)}{2ρ(n + 2)} + ρ(n + 2)\frac{v(n + 1)β(n + 1)}{2ρ(n + 1)})δ_{even}(n)δ_{m,n+2}

− (ρ(n)\frac{v(n − 1)β(n − 1)}{2ρ(n − 1)} + ρ(n − 1)\frac{v(n)β(n)}{2ρ(n)})δ_{odd}(n)δ_{m,n−2})_{m,n∈ℤ}.

□

Lemma 6.3. Assume Hypothesis 6.1. Then the following relations hold:

\frac{δĝ_{ℓ,+}}{δβ} = \frac{ℓ}{γ} (f̂_{ℓ−1,+} − αĝ_{ℓ,+}), ℓ ∈ ℕ,  (6.35)
\frac{δĝ_{ℓ,−}}{δβ} = −\frac{ℓ}{γ} (f̂_{ℓ−1,−}^{−} + αĝ_{ℓ,−}^{−}), ℓ ∈ ℕ.  (6.36)

44 F. GESZTESY, H. HOLDEN, J. MICHOR, AND G. TESCHL

Proof. We consider (6.35) first. By (3.21) one concludes that

\begin{matrix}f̂_{ℓ−1,+}(n) − α(n)ĝ_{ℓ,+}(n) \\ = (δ_{n}, EL^{ℓ−1}δ_{n})δ_{even}(n) + (δ_{n}, L^{ℓ−1}Dδ_{n})δ_{odd}(n) − α(n)(δ_{n}, L^{ℓ}δ_{n}). \end{matrix} (6.37)

Thus one has to show that

∑_{n∈ℤ}(δ_{n}, L^{ℓ−1}M_{β}δ_{n}) = ∑_{n∈ℤ} \frac{v(n)}{ρ(n)^{2}} (f̂_{ℓ−1,+}(n) − α(n)ĝ_{ℓ,+}(n)),  (6.38)

since this implies (6.35), using (6.24). By (6.30), (3.17), and (3.18), and assuming v ∈ ℓ^{1}(ℤ), one obtains

∑_{n∈ℤ}(δ_{n}, L^{ℓ−1}M_{β}δ_{n})

= ∑_{n∈ℤ} ( − vα^{+}(δ_{n}, L^{ℓ−1}δ_{n}) + v^{−}ρ(δ_{n}, L^{ℓ−1}δ_{n−1})δ_{odd} + vρ^{+}(δ_{n}, L^{ℓ−1}δ_{n+1})δ_{odd}

− \frac{vα}{2ρ}( − α^{+}(δ_{n}, L^{ℓ−1}δ_{n−1})δ_{even} + ρ^{+}(δ_{n+1}, L^{ℓ−1}δ_{n−1})δ_{even})

− \frac{vα}{2ρ}(β^{−}(δ_{n}, L^{ℓ−1}δ_{n−1})δ_{odd} + ρ^{−}(δ_{n}, L^{ℓ−1}δ_{n−2})δ_{odd})

− \frac{v^{+}α^{+}}{2ρ^{+}} ( − α^{++}(δ_{n}, L^{ℓ−1}δ_{n+1})δ_{even} + ρ^{++}(δ_{n}, L^{ℓ−1}δ_{n+1})δ_{even})

− \frac{v^{+}α^{+}}{2ρ^{+}} (β(δ_{n}, L^{ℓ−1}δ_{n+1})δ_{odd} + ρ(δ_{n−1}, L^{ℓ−1}δ_{n+1})δ_{odd}))

= ∑_{n∈ℤ} ( − vα^{+}(δ_{n}, L^{ℓ−1}δ_{n}) + vρ^{+}(δ_{n+1}, L^{ℓ−1}δ_{n})δ_{even} + vρ^{+}(δ_{n}, L^{ℓ−1}δ_{n+1})δ_{odd}

− \frac{vα}{2ρ}((δ_{n}, EL^{ℓ−1}δ_{n−1})δ_{even} + (δ_{n}, L^{ℓ−1}Dδ_{n−1})δ_{odd})

− \frac{v^{+}α^{+}}{2ρ^{+}} ((δ_{n}, L^{ℓ−1}Dδ_{n+1})δ_{even} + (δ_{n}, EL^{ℓ−1}δ_{n+1})δ_{odd}))

= ∑_{n∈ℤ} (v(δ_{n}, EL^{ℓ−1}δ_{n})δ_{even} + v(δ_{n}, L^{ℓ−1}Dδ_{n})δ_{odd}

− \frac{vα}{2ρ}((δ_{n}, EL^{ℓ−1}δ_{n−1})δ_{even} + (δ_{n}, L^{ℓ−1}Dδ_{n−1})δ_{odd}

+ (δ_{n−1}, L^{ℓ−1}Dδ_{n})δ_{odd} + (δ_{n−1}, EL^{ℓ−1}δ_{n})δ_{even}))

= ∑_{n∈ℤ} (\frac{v}{ρ^{2}} (f̂_{ℓ−1,+} − αĝ_{ℓ,+}) + \frac{vα}{2ρ}((δ_{n−1}, EL^{ℓ−1}δ_{n})δ_{even}  (6.39)

+ (δ_{n}, L^{ℓ−1}Dδ_{n−1})δ_{odd} − (δ_{n}, EL^{ℓ−1}δ_{n−1})δ_{even} − (δ_{n−1}, L^{ℓ−1}Dδ_{n})δ_{odd})), where we used (3.21) and

ĝ_{ℓ,+} = (δ_{n}, L^{ℓ−1}DEδ_{n})
= β(δ_{n}, L^{ℓ−1}Dδ_{n})δ_{odd} + ρ(δ_{n}, L^{ℓ−1}Dδ_{n−1})δ_{odd}
+ β(δ_{n}, EL^{ℓ−1}δ_{n})δ_{even} + ρ(δ_{n−1}, EL^{ℓ−1}δ_{n})δ_{even}
= βf̂_{ℓ−1,+} + ρ(δ_{n}, L^{ℓ−1}Dδ_{n−1})δ_{odd} + ρ(δ_{n−1}, EL^{ℓ−1}δ_{n})δ_{even}.  (6.40)

Hence it remains to show that (δ_{n−1}, EL^{ℓ−1}δ_{n})δ_{even} + (δ_{n}, L^{ℓ−1}Dδ_{n−1})δ_{odd}

− (δ_{n}, EL^{ℓ−1}δ_{n−1})δ_{even} − (δ_{n−1}, L^{ℓ−1}Dδ_{n})δ_{odd} = 0, (6.41)

but this follows from (EL^{ℓ})^{⊤} = EL^{ℓ} respectively (L^{ℓ}D)^{⊤} = L^{ℓ}D by (3.13), (3.16). In the case (6.36) one similarly shows that

∑_{n∈ℤ}(δ_{n}, L^{−ℓ−1}M_{β}δ_{n}) = − ∑_{n∈ℤ} \frac{v(n + 1)}{ρ(n)^{2}}((δ_{n}, D^{−1}L^{−ℓ+1}δ_{n})δ_{even}(n)  (6.42)

+ (δ_{n}, L^{−ℓ+1}E^{−1}δ_{n})δ_{odd}(n) + α(n + 1)(δ_{n}, L^{−ℓ}δ_{n})).

□

Lemma 6.4. Assume Hypothesis 6.1. Then the following relations hold:

\frac{δĝ_{ℓ,+}}{δα} = −\frac{ℓ}{γ} (ĥ_{ℓ−1,+}^{−} + βĝ_{ℓ,+}^{−}), ℓ ∈ ℕ,  (6.43)
\frac{δĝ_{ℓ,−}}{δα} = \frac{ℓ}{γ} (ĥ_{ℓ−1,−} − βĝ_{ℓ,−}), ℓ ∈ ℕ.  (6.44)

Proof. We consider (6.43) first. Using (6.25), (6.34), (3.17), and (3.18), and assuming v ∈ ℓ^{1}(ℤ) one obtains

∑_{n∈ℤ}(δ_{n}, L^{ℓ−1}M_{α}δ_{n})

= ∑_{n∈ℤ} ( − v^{+}β(δ_{n}, L^{ℓ−1}δ_{n}) − vρ(δ_{n}, L^{ℓ−1}δ_{n−1})δ_{even} − v^{++}ρ^{+}(δ_{n}, L^{ℓ−1}δ_{n+1})δ_{even}

− \frac{vβ}{2ρ} (β^{−}(δ_{n}, L^{ℓ−1}δ_{n−1})δ_{odd} + ρ^{−}(δ_{n}, L^{ℓ−1}δ_{n−2})δ_{odd})

− \frac{vβ}{2ρ} ( − α^{+}(δ_{n}, L^{ℓ−1}δ_{n−1})δ_{even} + ρ^{+}(δ_{n+1}, L^{ℓ−1}δ_{n−1})δ_{even})

− \frac{v^{+}β^{+}}{2ρ^{+}} ( − α^{++}(δ_{n}, L^{ℓ−1}δ_{n+1})δ_{even} + ρ^{++}(δ_{n}, L^{ℓ−1}δ_{n+1})δ_{even})

− \frac{v^{+}α^{+}}{2ρ^{+}} (β(δ_{n}, L^{ℓ−1}δ_{n+1})δ_{odd} + ρ(δ_{n−1}, L^{ℓ−1}δ_{n+1})δ_{odd}))

= ∑_{n∈ℤ} ( − v^{+}((δ_{n}, L^{ℓ−1}Dδ_{n})δ_{even} + (δ_{n}, EL^{ℓ−1}δ_{n})δ_{odd})

− \frac{vβ}{2ρ} ((δ_{n}, EL^{ℓ−1}δ_{n−1})δ_{even} + (δ_{n}, L^{ℓ−1}Dδ_{n−1})δ_{odd})

− \frac{v^{+}α^{+}}{2ρ^{+}} ((δ_{n}, L^{ℓ−1}Dδ_{n+1})δ_{even} + (δ_{n}, EL^{ℓ−1}δ_{n+1})δ_{odd}))

= − ∑_{n∈ℤ} \frac{v}{ρ^{2}} (ĥ_{ℓ−1,+}^{−} + βĝ_{ℓ,+}^{−}),  (6.45)

since by (3.21) and (5.9),

2αĥ_{ℓ−1,+}^{−} + 2ĝ_{ℓ,+}^{−} = ρ((δ_{n}, L^{ℓ−1}Dδ_{n−1})δ_{odd} + (δ_{n}, EL^{ℓ−1}δ_{n−1})δ_{even}
+ (δ_{n−1}, L^{ℓ−1}Dδ_{n})δ_{odd} + (δ_{n−1}, EL^{ℓ−1}Dδ_{n})δ_{even}).

(6.46)

The result (6.44) follows similarly. □

46 F. GESZTESY, H. HOLDEN, J. MICHOR, AND G. TESCHL

Next, we introduce the Hamiltonians

Ĥ_{0} = ∑_{n∈ℤ} ln(γ(n)),  Ĥ_{p_{±},±} = \frac{1}{p_{±}} ∑_{n∈ℤ} ĝ_{p_{±},±}(n), p_{±} ∈ ℕ,  (6.47)
H\underline{p} = ∑_{ℓ=1}^{p_{+}} c_{p_{+}−ℓ,+}Ĥ_{ℓ,+} + ∑_{ℓ=1}^{p_{−}} c_{p_{−}−ℓ,−}Ĥ_{ℓ,−} + c\underline{p}Ĥ_{0}, \underline{p} = (p_{−}, p_{+}) ∈ ℕ_{0}^{2}. (6.48)

(We recall that c\underline{p} = (c_{p,−} + c_{p,+})/2.) Theorem 6.5. Assume Hypothesis 6.1. Then the following relations hold:

AL\underline{p}(α, β) = (\begin{matrix}−iα_{t\underline{p}} \\ −iβ_{t\underline{p}} \end{matrix}) + D∇H\underline{p} = 0, \underline{p} ∈ ℕ_{0}^{2}.  (6.49)

Proof. This follows directly from Lemmas 6.3 and 6.4,

(∇Ĥ_{ℓ,+})_{α} = \frac{1}{γ} ( − βĝ_{ℓ,+}^{−} − ĥ_{ℓ−1,+}^{−}), (∇Ĥ_{ℓ,+})_{β} = \frac{1}{γ} ( − αĝ_{ℓ,+} + f̂_{ℓ−1,+}),
(∇Ĥ_{ℓ,−})_{α} = \frac{1}{γ} ( − βĝ_{ℓ,−} + ĥ_{ℓ−1,−}), (∇Ĥ_{ℓ,−})_{β} = \frac{1}{γ} ( − αĝ_{ℓ,−}^{−} − f̂_{ℓ−1,−}^{−}), (6.50)
ℓ ∈ ℕ,

together with (2.19). □ Theorem 6.6. Assume Hypothesis 5.1 and suppose that α, β satisfy AL\underline{p}(α, β) = 0 for some \underline{p} ∈ ℕ_{0}^{2}. Then,

\frac{dH\underline{r}}{dt\underline{p}} = 0, \underline{r} ∈ ℕ_{0}^{2}.  (6.51)

Proof. From Lemma 4.4 and Theorem 5.7 one obtains

\frac{dĝ_{r_{±},±}}{dt\underline{p}} = (S^{+} − I)J_{r_{±},±}, r_{±} ∈ ℕ_{0},  (6.52)

for some J_{r_{±},±}, r_{±} ∈ ℕ_{0}, which are polynomials in α and β and certain shifts thereof. Using definition (6.48) of H\underline{r}, the result (6.51) follows in the homogeneous case and then by linearity in the general case. □ Theorem 6.7. Assume Hypothesis 6.1 and let \underline{p}, \underline{r} ∈ ℕ_{0}^{2}. Then,

\{H\underline{p}, H\underline{r}\} = 0,  (6.53)

that is, H\underline{p} and H\underline{r} are in involution for all \underline{p}, \underline{r} ∈ ℕ_{0}^{2}. Proof. By Theorem 5.5, there exists T > 0 such that the initial value problem

AL\underline{p}(α, β) = 0, (α, β)|_{t\underline{p}=0} = (α^{(0)}, β^{(0)}),  (6.54)

where α^{(0)}, β^{(0)} satisfy Hypothesis 6.1, has unique, local, and smooth solutions α(t), β(t) satisfying Hypothesis 6.1 for each t ∈ [0, T). For this solution we know that

\frac{d}{dt\underline{p}} H\underline{p}(t) = \{H\underline{r}(t), H\underline{p}(t)\} = 0.  (6.55)

Next, let t ↓ 0. Then

0 = \{H\underline{r}(t), H\underline{p}(t)\} →_{t↓0} \{H\underline{r}(0), H\underline{p}(0)\} = \{H\underline{r}, H\underline{p}\}|_{(α,β)=(α^{(0)},β^{(0)})}.  (6.56)

Since α^{(0)}, β^{(0)} are arbitrary coefficients satisfying Hypothesis 6.1 one concludes (6.53). □

Acknowledgments. Fritz Gesztesy, Johanna Michor, and Gerald Teschl gratefully acknowledge the extraordinary hospitality of the Department of Mathematical Sciences of the Norwegian University of Science and Technology, Trondheim, during extended stays in the summers of 2006 and 2007, where parts of this paper were written.

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Department of Mathematics, University of Missouri, Columbia, MO 65211, USA

E-mail address: fritz@math.missouri.edu
URL: http://www.math.missouri.edu/personnel/faculty/gesztesyf.html

Department of Mathematical Sciences, Norwegian University of Science and Technology, NO–7491 Trondheim, Norway

E-mail address: holden@math.ntnu.no
URL: http://www.math.ntnu.no/~holden/

Department of Mathematics, Imperial College, 180 Queen’s Gate, London SW7 2BZ, and International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Wien, Austria

E-mail address: Johanna.Michor@esi.ac.at
URL: http://www.mat.univie.ac.at/~jmichor/

Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090 Wien, Austria, and International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Wien, Austria

E-mail address: Gerald.Teschl@univie.ac.at
URL: http://www.mat.univie.ac.at/~gerald/