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arXiv math-ph/0211015math-phmath.MPmath.SP
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具有不定号位势的离散Schr dinger算子的变分估计

Variational Estimates for Discrete Schrödinger Operators with Potentials of Indefinite Sign

David Damanik, Dirk Hundertmark, Rowan Killip, Barry Simon

备注：
17 pages
发表信息：
Commun. Math. Phys. 238 (2003), 545-562
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摘要：

设$H$是一维离散薛定谔算子。证明了如果$σ_{\ess} (H)\subset [-2,2]$，则$H-H_0$是紧的，并且是$σ_{\ess}(H)=[-2,2]$ .我们还证明了，如果$H_0 + \frac14 V^2$至少有一个束缚态，那么同样的情况也适用于$H_0 +V$。此外，如果$H_0 + \frac14 V^2$有无穷多个束缚态，则$H_0 +V$也有无穷多个束缚态。结果包括这样一个事实，即对于带有$\liminf_{|n|\to\infty} |nV(n)| > 1$的衰变势$V$，$H_0 +V$具有无穷多个束缚态; $V$的符号是无关紧要的。高维类似物也进行了讨论。

英文摘要：

Let $H$ be a one-dimensional discrete Schrödinger operator. We prove that if $σ_{\ess} (H)\subset [-2,2]$, then $H-H_0$ is compact and $σ_{\ess}(H)=[-2,2]$. 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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