Semi-classical spectral estimates for Schrödinger operators at a critical level. Case of a degenerate maximum of the potential

Brice Camus

doi:10.1016/j.jde.2005.10.003

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Semi-classical spectral estimates for Schrödinger operators at a critical level. Case of a degenerate maximum of the potential

Brice Camus

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Abstract

We study the semi-classical trace formula at a critical energy level for a Schrödinger operator on $\mathbb{R}^{n}$. We assume here that the potential has a totally degenerate critical point associated to a local maximum. The main result, which establishes the contribution of the associated equilibrium in the trace formula, is valid for all time in a compact subset of $\mathbb{R}$ and includes the singularity in $t=0$. For these new contributions the asymptotic expansion involves the logarithm of the parameter $h$. Depending on an explicit arithmetic condition on the dimension and the order of the critical point, this logarithmic contribution can appear in the leading term.

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BibTeX RIS

Brice Camus. 2005-02-27. Semi-classical spectral estimates for Schrödinger operators at a critical level. Case of a degenerate maximum of the potential. https://doi.org/10.1016/j.jde.2005.10.003

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