Comparison theorems for eigenvalues of one-dimensional Schrödinger operators

Abstract

The Schrödinger operator $H = -d^{2}/dx^{2}+V (x)$ on an interval $[0, a]$ with Dirichlet or Neumann boundary conditions has discrete spectrum $E_{1}[V] < E_{2}[V] < E_{3}[V] < \cdots$, for bounded $V$. In this paper, we apply the perturbation theory of discrete eigenvalues to obtain upper bounds for $\sum_{j=1}^{k} E_{j}[V]$, where $k$ is any positive integer. Our results include the following:

(i) $\sum_{j=1}^{k} E_{j}[V]\leq \sum_{j=1}^{k} E_{j}[V_{s}]$, where $V_{s}(x) = [V (x)+V (a-x)]/2$, with equality if and only if $V$ is symmetric about $x = a/2$.

(ii) If $V$ is convex, then the Dirichlet eigenvalues satisfy \[ \sum_{j=1}^{k} E_{j}[V]\leq \sum_{j=1}^{k} E_{j}[0]+\frac{k}{a}\int _{0}^{a}V (x)dx \] with equality if and only if V is constant.

(iii) If $V$ is concave, then the Neumann eigenvalues satisfy \[ \sum_{j=1}^{k} E_{j}[V]\leq \sum_{j=1}^{k} E_{j}[0]+\frac{k}{a}\int _{0}^{a}V (x)dx \] with equality if and only if $V$ is constant.