Mathematics – Spectral Theory
Scientific paper
2005-02-27
Journal of Differential Equations. Volume 226, Issue 1 , 1 July 2006, Pages 295-322
Mathematics
Spectral Theory
27 pages, perhaps to be revised
Scientific paper
10.1016/j.jde.2005.10.003
We study the semi-classical trace formula at a critical energy level for a Schr\"odinger 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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