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

Consider the Schrödinger equation $-y'' + Vy = \lambda y$ for a potential $V$ of period 1 in the weighted Sobolev space $(N \in \mathbb{Z}_{\ge 0}, \omega \in \mathbb{R}_{\ge 0})$

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

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)

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