93-244 R. Brummelhuis, T. Paul, A. Uribe.

Spectral Estimates Around a Critical Level (141K, LaTeX) Oct 8, 93

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Abstract. We study the semi-classical 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 non-degeneracy assumptions we obtain an asymptotic expansion of the sum $\sum_j\varphi((E_j-E_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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