Variational Estimates for Discrete Schrödinger Operators with Potentials of Indefinite Sign

Abstract

Let $H$ be a one-dimensional discrete Schrödinger operator. We prove that if $σ_{\ess} (H)\subset [-2,2]$, then $H-H_0$ is compact and $σ_{\ess}(H)=[-2,2]$. We also prove that if $H_0+\frac{1}{4}{V}^2$ has at least one bound state, then the same is true for $H_0+V$. Further, if $H_0+\frac{1}{4}{V}^2$ has infinitely many bound states, then so does $H_0+V$. Consequences include the fact that for decaying potential $V$ with ${\mathrm{liminf}}_{|n|\to \infty }|nV(n)|>1$, $H_0+V$ has infinitely many bound states; the signs of $V$ are irrelevant. Higher-dimensional analogues are also discussed.

17 pages

References in corpus (2)

• Sum Rules and the Szego Condition for Orthogonal Polynomials on the Real Line
• Half-line Schrodinger Operators With No Bound States

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