Semiclassical Resonances of Schrödinger operators as zeroes of regularized determinants

Details Semiclassical Resonances of Schrödinger operators as zeroes of regularized determinants Semiclassical Resonances of Schrödinger operators as zeroes of regularized determinants

2007-09-13

arxiv.org/abs/0709.2060v3

International Mathematics Research Notices 2008 (2008) ID rnn002, 55 pages

Mathematics

Spectral Theory

37 pages, published version

Scientific paper

10.1093/irmn/rnn002

We prove that the resonances of long range perturbations of the (semiclassical) Laplacian are the zeroes of natural perturbation determinants. We more precisely obtain factorizations of these determinants of the form $ \prod_{w = {\rm resonances}}(z-w) \exp (\varphi_p(z,h)) $ and give semiclassical bounds on $ \partial_z \varphi_p $ as well as a representation of Koplienko's regularized spectral shift function. Here the index $ p \geq 1 $ depends on the decay rate at infinity of the perturbation.

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