Universal lower bounds on eigenvalue splittings for one dimensional Schrödinger operators

Abstract

We provide lower bounds on the eigenvalue splitting for −d ²/dx ²+V(x) depending only on qualitative properties ofV. For example, ifV is C^∝ on [a, b] andE _n ,E _n−1 are two successive eigenvalues of −d ²/dx ²+V withu(a)=u(b)=0 boundary conditions, and if$\lambda = \mathop {\max }\limits_{E \in (E_{n - 1} ,E_n );x \in (a,b)} |E - V(x)|^{1/2}$, then

$$E_n - E_{n - 1} \geqq \pi \lambda ^2 \exp \left[ { - \lambda (b - a)} \right]$$

. The exponential factor in such bounds are saturated precisely in tunneling examples. Our results arenot restricted toV's of compact support, but only require$E_n< \mathop {\lim }\limits_{\overline {x \to \infty } } V(x)$.