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

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
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### References:

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