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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 non-selfadjoint, eigenvalue, periodic flow, branching singularity
MSC Classifications: 31C10, 35P20, 35Q40, 37J35, 37J45, 53D22, 58J40 show english descriptions Pluriharmonic and plurisubharmonic functions [See also 32U05]
Asymptotic distribution of eigenvalues and eigenfunctions
PDEs in connection with quantum mechanics
Completely integrable systems, topological structure of phase space, integration methods
Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
Canonical transformations
Pseudodifferential and Fourier integral operators on manifolds [See also 35Sxx]
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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