 02451 David Damanik, Dirk Hundertmark, Rowan Killip, and Barry Simon
 Variational Estimates for Discrete Schr\"odinger Operators with Potentials of Indefinite Sign
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Nov 10, 02

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Abstract. Let $H$ be a onedimensional discrete Schr\"odinger operator. We prove that if $\sigma_{\ess} (H)\subset [2,2]$, then $HH_0$ 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. Higherdimensional analogues are also discussed.
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