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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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.