Sharp semiclassical estimates for the number of eigenvalues below a degenerate critical level

Details Sharp semiclassical estimates for the number of eigenvalues below a degenerate critical level Sharp semiclassical estimates for the number of eigenvalues below a degenerate critical level

2007-02-22

arxiv.org/abs/math/0702665v1

Mathematics

Spectral Theory

33 pages

Scientific paper

We consider the semiclassical asymptotic behaviour of the number of eigenvalues smaller than $E$ for elliptic operators in $L\sp 2 ({\bf R}\sp d)$. We describe a method of finding remainder estimates related to the volume of the region of the phase space in which the principal symbol takes values belonging to the interval $[E'-h; E'+h]$, where $E'$ is close to $E$. This method allows to derive remainder estimates $O(h\sp {1-d})$ for a class of symbols with critical points and non-smooth coefficients.

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