Evgeni Korotyaev
The Second Order Estimates for the Hill Operator
Preprint series: ESI preprints

MSC:

34B30 Special equations (Mathieu, Hill, Bessel, etc.)

Abstract: Let $H$ be the Hill operator and
let $G_n=(A_n^-, A_n^+)$ and $M_n^{\pm}, n\geq 1,$
be the corresponding gaps and the effective masses for the Hill operator.
Denote by $F(E)$ the Lyapunov function for the Hill operator.
Let $\lambda _n \in [A_n^-, A_n^+]$ be such that $F'(\lambda _n)=0$ and
$h_n\geq 0$ be the solution of the equation
$\cosh h_n=(-1)^nF(\lambda _n).$
We prove some identities for the effective masses $M_n^{\pm}.$
Then we find "second order estimates" with respect to $|G_n|$
for $\lambda _n, h_n,
M_n^{\pm} $ (for example $|\lambda _n-\frac{1}{2}(A_n^-+ A_n^+)|
\leq C|G_n|^2/n^2 $),
and we get ones for more general cases
(the Dirac operator with periodic coefficients etc.).

Keywords: Hill operator, effective mass, Lyapunov function