Variational Estimates for Discrete Schrödinger Operators with Potentials of Indefinite Sign

Details Variational Estimates for Discrete Schrödinger Operators with Potentials of Indefinite Sign Variational Estimates for Discrete Schrödinger Operators with Potentials of Indefinite Sign

2002-11-11

arxiv.org/abs/math-ph/0211015v1

Physics

Mathematical Physics

17 pages

Scientific paper

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.

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