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)⊂[−2,2], then H−H_0 is compact and Σ_(ess)(H)=[−2,2]. We also prove that if H_0 + 1/4 V^2 has at least one bound state, then the same is true for H_0 + V. Further, if H_0 + 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 lim inf_(|n|→ ∞|nV(n)|> 1 lim inf_(|n|→∞)|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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