Mathematics Colloquia and Seminars

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Description

Variational bounds have been one of the major tools in the study of discrete eigenvalues of Schr\"odinger operators. However, they are usually not applicable if the potential is allowed to change signs. We give a simple variational bound for eigenvalues of discrete Schr\"odinger operators which does not require the potential to be of fixed sign. A consequence of this bound is a simple proof of the fact that if the spectrum of a discrete Schr\"odinger operator on $\mathbb{Z}$ is [-2,2], then the potential must vanish identically. Also, certain eigenvalue moments must blow up for slowly decaying potentials, even if they are highly oscillating. Most results extend to higher dimensions.