 93244 R. Brummelhuis, T. Paul, A. Uribe.
 Spectral Estimates Around a Critical Level
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Oct 8, 93

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Abstract. We study the semiclassical distribution of the eigenvalues of a
Schrodinger operator in a neighborhood of size $\hbar$ of a critical
value of the classical Hamiltonian. More precisely, under certain
nondegeneracy assumptions we obtain an asymptotic expansion of the
sum $\sum_j\varphi((E_jE_c)/\hbar) where $E_j$ are the eigenvalues,
$E_c$ the critical energy level and $\varphi$ is a test function with
compactly supported Fourier transform. In addition to powers of
$1/\hbar$, the expansion in general contains logarithmic terms. We
compute the coefficient of the greatest such term from the classical
Hamilton flow of the system. For the double well in one degree of
freedom, this gives a logarithmic Weyl law for the number of
eigenvalues in $[E_c\hbar, E_c+\hbar]$ where $E_c$ is the local
maximum of the potential. We also obtain estimates on the
eigenfunctions.
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