Comparison theorems for eigenvalues of one-dimensional Schrödinger operators (Q1890288)

This paper deals with estimates on the sum of low-lying eigenvalues of one-dimensional Schrödin\-ger operators. Let \(\{V(\cdot,t)\}_{t\in\mathbb{R}}\) be a family of bounded, real-valued functions on \([0,a]\). For \(j\in\mathbb{N}\), we denote by \(E_{j}(t)\) the \(j\)th eigenvalue of the operator \(H(t)=-d^{2}/dx^{2}+V(x,t)\) in \(L^{2}((0,a))\) with Dirichlet or Neumann boundary conditions. The main results of this paper are the following: (1) If \(H(t)\) extends to an analytic family of type \((A)\) and if \((\partial^{2}V/\partial t^{2})(x,t)\leq 0\), then\break \((d^{2}/dt^{2})\sum^{k}_{j=1}E_{j}(t)\leq 0\) for any \(k\geq 1\). (2) If \(V(x,t)\) is concave with respect to \(t\), then \(\sum^{k}_{j=1}E_{j}(t)\) is a concave function of \(t\) for any \(k\geq 1\). As applications of the above results, the author obtains the following optimal estimates. Let \(E_{j}[V]\) be the \(j\)th eigenvalue of the operator \(H\) in \(L^{2}((0,a))\) with Dirichlet or Neumann boundary conditions, where \(V\) is a bounded, real-valued function on \((0,a)\). Then it holds true: (i) \(\sum^{k}_{j=1}E_{j}[V]\leq\sum^{k}_{j=1}E_{j}[V_{s}]\), 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 \[ \sum^{k}_{j=1}E_{j}(V)\leq\sum^{k}_{j=1}E_{j}[0]+\frac{k}{a}\int^{a}_{0}V(x)\,dx \] with equality if and only if \(V\) is constant. (iii) If \(V\) is concave, then the Neumann eigenvalues satisfy \[ \sum^{k}_{j=1}E_{j}(V)\leq\sum^{k}_{j=1}E_{j}[0]+\frac{k}{a}\int^{a}_{0}V(x)\,dx \] with equality if and only if \(V\) is constant.