## Gap estimates of the spectrum of Hill’s equation and action variables for KdV.(English)Zbl 0924.58074

Summary: Consider the Schrödinger equation $$-y'' + Vy = \lambda y$$ for a potential $$V$$ of period 1 in the weighted Sobolev space $$(N \in \mathbb{Z}_{\geq 0}, \;\omega \in \mathbb{R}_{\geq 0})$$ $H^{N, \omega}(S^1; \mathbb{C}) :=\left \{ f(x) = \sum^{\infty}_{k= - \infty} \;\Hat{\Hat f}(k) e^{i 2 \pi kx} \;\bigg | \| f \|_{N, \omega} < \infty\right \}$ where $$\Hat{\Hat f}(k) \;(k \in \mathbb{Z})$$ denote the Fourier coefficients of $$f$$ when considered as a function of period 1, $\| f \|_{N, \omega} \;:= \;\bigg( \sum_k (1+| k|)^{2N} e^{2 \omega | k | } | \;\;\Hat{\Hat{f}} (k) | ^2 \bigg)^{1/2} < \infty ,$ and where $$S^1$$ is the circle of length 1. Denote by $$\lambda_k \equiv \lambda_k (V) \;(k \geq 0)$$ the periodic eigenvalues of $$- \frac{d^2}{dx^2} + V$$ when considered on the interval $$[0,2],$$ with multiplicities and ordered so that Re$$\lambda_j \leq\text{Re} \lambda_{j+1} \;(j \geq 0).$$ We prove the following result.
Theorem. For any bounded set $${\mathcal B} \subseteq H^{N, \omega} (S^1; \mathbb{C}),$$ there exist $$n_0 \geq 1$$ and $$M \geq 1$$ so that for $$k \geq n_0$$ 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) $$\sum_{k \geq n_0} (1+k)^{2N} e^{2 \omega k} | \lambda_k^+ - \lambda^-_k | ^2 \leq M$$,
(ii) $$\sum_{k \geq n_0} (1 + k)^{2 N+1} e^{2 \omega k} \bigg | (\lambda^+_k - \lambda^-_k) -2 \sqrt{\Hat{\Hat{V}} (k) \Hat{\Hat{V}}(-k)} \bigg | ^2 \leq M$$.

### MSC:

 37A30 Ergodic theorems, spectral theory, Markov operators 35Q35 PDEs in connection with fluid mechanics 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 37N99 Applications of dynamical systems
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### References:

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