On the location of resonances for Schrödinger operators in the semiclassical limit. II: Barrier top resonances. (English) Zbl 0622.47047

Briet, P.; Combes, J. M.; Duclos, P.

Commun. Partial Differ. Equations 12, 201-222 (1987).

It is shown that for a Schrödinger operator \(H=-\hslash^ 2\Delta +V\) on \(L^ 2({\mathbb{R}}^ n)\) and in the semi-classical regime where \(\hslash\) is small, an absolute nondegenerate maximum of the potential at \(x=0\) with energy \(E_ 0\), creates resonances. The resonances energies \(E_ n\) have asymptotic expansions and to first order: \(E_ n=E_ 0- i\hslash e_ n+O(\hslash^{3/2})\) where the energies \(e_ n\) are the eigenvalues of the harmonic oscillator: \(K=-\Delta +x Ax\) with \(A_{ij}=-(1/2)\partial^ 2V/\partial x_ i\partial x_ j|_{x=0}.\)
Such resonances correspond to classically trapped particles at energy \(E_ 0\). This result has been obtained simultaneously by J. Sjöstrand who also consideres the case where the matrix A can be degenerate. [Part I is to appear in J. Math. Anal. Appl.]

MSC:

47F05	General theory of partial differential operators
81Q05	Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
47A10	Spectrum, resolvent

Keywords:

Schrödinger operator; semi-classical regime; resonances energies; asymptotic expansions; eigenvalues of the harmonic oscillator; trapped particles

References:

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