Sharp spectral estimates in domains of infinite volume

Details Sharp spectral estimates in domains of infinite volume Sharp spectral estimates in domains of infinite volume

2010-12-29

arxiv.org/abs/1012.5972v1

Rev. Math. Phys. 23 (2011), no. 6, 615-641

Mathematics

Spectral Theory

21 pages, 2 figures

Scientific paper

We consider the Dirichlet Laplace operator on open, quasi-bounded domains of infinite volume. For such domains semiclassical spectral estimates based on the phase-space volume - and therefore on the volume of the domain - must fail. Here we present a method how one can nevertheless prove uniform bounds on eigenvalues and eigenvalue means which are sharp in the semiclassical limit. We give examples in horn-shaped regions and so-called spiny urchins. Some results are extended to Schr\"odinger operators defined on quasi-bounded domains with Dirichlet boundary conditions.

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