Estimates for Periodic and Dirichlet Eigenvalues of the Schrödinger Operator

Abstract

Consider the Schrödinger equation $-y' + Vy = \lambda y$ for a complex-valued potential V of period 1 in the weighted Sobolev space $H^w$ of 2-periodic functions $f : \mathbb R \rightarrow \mathbb C$, \[ H^w \equiv H^w_{\mathbb C}:= \left\{ f(x) = \sum^\infty_{k= -\infty } \hat f(k) e^{i\pi kx} | \; \| f\|_w < \infty \right\}, \] where \[ \| f\|_w:= \left(2\sum_k w(k)^2 \ |{\hat f}(k)|^2\right)^{1/2} \] and $w = (w(k))_{k\in {\mathbb Z}}$ denotes a symmetric, submultiplicative weight sequence. Denote by $\lambda_n = \lambda_n(V) \ (n \geq 0)$ the periodic eigenvalues of $- \frac {d^2}{dx^2} + V$ when considered on the interval $[0,2]$, listed in such a way that $\lambda_{2n}, \lambda_{2n- 1} = n^2 \pi^2 + 0(1)$, and denote by $\mu_n = \mu_n(V) \ (n \geq 1)$ the Dirichlet eigenvalues of $- \frac {d^2}{dx^2} + V$ considered on $[0,1]$, listed in such a way that $\mu_n = n^2 \pi^2 + 0(1)$. {\sc Theorem.} {\it There exist (absolute) constants $K_1,K_2 > 0$, so that for any 1-periodic potential V in $H^w$, \[ \sum_{n\geq N} w (2n)^2 |\lambda_{2n} - \lambda_{2n-1}|^2 \leq K_1(1 + \| V\|_w)^{K_2} \] and \[ \sum_{n\geq N}w(2n)^2 | \mu_n - \lambda_{2n} |^2 \leq K_1 (1 + \| V\|_w)^{K_2} , \] where $N:= K_1(1 + \| V\|_w)^2$.} \\