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# Non-Selfadjoint Perturbations of Selfadjoint Operators in Two Dimensions IIIa. One Branching Point

Published:2008-06-01
Printed: Jun 2008
• Michael Hitrik
• Johannes Sj{östrand
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## Abstract

This is the third in a series of works devoted to spectral asymptotics for non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2, having a periodic classical flow. Assuming that the strength $\epsilon$ of the perturbation is in the range $h^2\ll \epsilon \ll h^{1/2}$ (and may sometimes reach even smaller values), we get an asymptotic description of the eigenvalues in rectangles $[-1/C,1/C]+i\epsilon [F_0-1/C,F_0+1/C]$, $C\gg 1$, when $\epsilon F_0$ is a saddle point value of the flow average of the leading perturbation.
 Keywords: non-selfadjoint, eigenvalue, periodic flow, branching singularity
 MSC Classifications: 31C10 - Pluriharmonic and plurisubharmonic functions [See also 32U05] 35P20 - Asymptotic distribution of eigenvalues and eigenfunctions 35Q40 - PDEs in connection with quantum mechanics 37J35 - Completely integrable systems, topological structure of phase space, integration methods 37J45 - Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods 53D22 - Canonical transformations 58J40 - Pseudodifferential and Fourier integral operators on manifolds [See also 35Sxx]

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