Variational Estimates for Discrete Schr\"odinger Operators with Potentials of Indefinite Sign

TL;DR

This paper establishes conditions under which discrete Schr"odinger operators have their essential spectrum preserved and bound states characterized, even with potentials of indefinite sign, extending to higher dimensions.

Contribution

It introduces variational estimates that relate the spectral properties of Schr"odinger operators with indefinite potentials to those with non-negative potentials, including higher-dimensional cases.

Findings

If the essential spectrum is within [-2,2], then the operator differs from the free operator by a compact perturbation.

Presence of a bound state in a modified operator implies the same in the original operator.

Potentials decaying faster than 1/|n| with certain bounds lead to infinitely many bound states.

Abstract

Let $H$ be a one-dimensional discrete Schr\"odinger operator. We prove that if $\sigma_{\ess} (H)\subset [-2,2]$, then $H-H_0$ is compact and $\sigma_{\ess}(H)=[-2,2]$. We also prove that if $H_0 + \frac14 V^2$ has at least one bound state, then the same is true for $H_0 +V$. Further, if $H_0 + \frac14 V^2$ has infinitely many bound states, then so does $H_0 +V$. Consequences include the fact that for decaying potential $V$ with $\liminf_{|n|\to\infty} |nV(n)| > 1$, $H_0 +V$ has infinitely many bound states; the signs of $V$ are irrelevant. Higher-dimensional analogues are also discussed.