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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Gap estimates
of the spectrum of Hill's equation
and action variables for $\mathbf{KdV}$


Authors: T. Kappeler and B. Mityagin
Journal: Trans. Amer. Math. Soc. 351 (1999), 619-646
MSC (1991): Primary 58F19, 58F07, 35Q35
MathSciNet review: 1473448
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Abstract: Consider the Schrödinger equation $-y'' + Vy = \lambda y$ for a potential $V$ of period 1 in the weighted Sobolev space $(N \in \mathbb{Z}_{\ge 0}, \ \omega \in \mathbb{R}_{\ge 0})$

\begin{displaymath}H^{N, \omega}(S^1; \mathbb{C}) := \{ f(x) = \sum^{\infty}_{k= - \infty} \ \Hat {\Hat f}(k) e^{i 2 \pi kx} \ \bigg | \parallel f \parallel _{N, \omega} < \infty \}\end{displaymath}

where $\Hat {\Hat f}(k) \ (k \in \mathbb{Z})$ denote the Fourier coefficients of $f$ when considered as a function of period 1,

\begin{displaymath}\parallel f \parallel _{N, \omega} \ := \ \bigg ( \sum _k (1+| k|)^{2N} e^{2 \omega | k |} | \ \ \Hat{\Hat{f}} (k) |^2 \bigg )^{^{1}/2} < \infty ,\end{displaymath}

and where $S^1$ is the circle of length 1. Denote by $\lambda _k \equiv \lambda _k (V) \ (k \ge 0)$ the periodic eigenvalues of $ - \frac{d^2}{dx^2} + V$ when considered on the interval $[0,2],$ with multiplicities and ordered so that $Re \lambda _j \le Re \lambda _{j+1} \ (j \ge 0).$ We prove the following result.

Theorem. For any bounded set ${\mathcal B} \subseteq H^{N, \omega} (S^1; \mathbb{C}),$ there exist $n_0 \ge 1$ and $M \ge 1$ so that for $k \ge n_0$ and $V \in {\mathcal B}$, the eigenvalues $\lambda _{2k}, \lambda _{2k -1}$ are isolated pairs, satisfying (with $\{ \lambda _{2k}, \lambda _{2k-1} \} = \{ \lambda^+_k , \lambda^-_k \})$

(i)
$\sum _{k \ge n_0} (1+k)^{2N} e^{2 \omega k} | \lambda _k^+ - \lambda^-_k |^2 \le M$,
(ii)
$\sum _{k \ge n_0} (1 + k)^{2 N+1} e^{2 \omega k} \bigg | (\lambda^+_k - \lambda^-_k) -2 \sqrt{\Hat{\Hat{V}} (k) \Hat{\Hat{V}}(-k)} \bigg |^2 \le M$.


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Additional Information

T. Kappeler
Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
Email: tk@math.unizh.ch

B. Mityagin
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210
Email: borismit@math.ohio-state.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02186-8
PII: S 0002-9947(99)02186-8
Received by editor(s): December 5, 1996
Article copyright: © Copyright 1999 American Mathematical Society