J. Spectr. Theory 4, 715-768 (2014) [DOI: 10.4171/JST/84]
Supersymmetry and Schrödinger-Type Operators with Distributional Matrix-Valued Potentials
Building on work on Miura's transformation by Kappeler, Perry, Shubin, and Topalov, we develop a detailed spectral theoretic treatment of Schrödinger operators with matrix-valued potentials, with special emphasis on distributional potential coefficients.
Here A= Im (d/dx) + φ in L2(ℝ)m, with a
matrix-valued coefficient φ = φ* ∈ L1loc(ℝ)m × m, m ∈ ℕ,
thus explicitly permitting distributional potential coefficients Vj in Hj, j=1,2,
in L2(ℝ)2m and H1 = A* A,
H2 = A A* in L2(ℝ)m.
Hj = - Im d2/dx2 + Vj(x), Vj(x) = φ(x)2 + (-1)j φ'(x), j=1,2.
Upon developing Weyl-Titchmarsh theory for these generalized Schrödinger operators Hj,
with (possibly, distributional) matrix-valued potentials Vj, we provide some spectral
theoretic applications, including a derivation of the corresponding spectral representations
for Hj, j=1,2. Finally, we derive a local Borg-Marchenko uniqueness theorem for Hj,
j=1,2, by employing the underlying supersymmetric structure and reducing it to the known
local Borg-Marchenko uniqueness theorem for D.
Our principal method relies on a supersymmetric (factorization) formalism underlying Miura's transformation, which intimately connects the triple of operators (D, H1, H2) of the form