Gap estimates of the spectrum of Hill’s equation and action variables for KdV

Abstract:

Consider the Schrödinger equation $-y'' + Vy = \lambda y$ for a potential $V$ of period 1 in the weighted Sobolev space $(N \in \mathbb {Z}_{\ge 0}, \omega \in \mathbb {R}_{\ge 0})$ \[ H^{N, \omega }(S^1; \mathbb {C}) := \{ f(x) = \sum ^{\infty }_{k= - \infty } \hat {\hat f}(k) e^{i 2 \pi kx} \bigg | \parallel f \parallel _{N, \omega } < \infty \}\] where $\hat {\hat f}(k) (k \in \mathbb {Z})$ denote the Fourier coefficients of $f$ when considered as a function of period 1, \[ \parallel f \parallel _{N, \omega } := \bigg ( \sum _k (1+| k|)^{2N} e^{2 \omega | k |} | \hat {\hat {f}} (k) |^2 \bigg )^{^{1}/2} < \infty ,\] and where $S^1$ is the circle of length 1. Denote by $\lambda _k \equiv \lambda _k (V) (k \ge 0)$ 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)] $\sum _{k \ge n_0} (1 + k)^{2 N+1} e^{2 \omega k} \bigg | (\lambda ^+_k - \lambda ^-_k) -2 \sqrt {\hat {\hat {V}} (k) \hat {\hat {V}}(-k)} \bigg |^2 \le M$.