Non-selfadjoint perturbations of selfadjoint operators in 2 dimensions. I. (English) Zbl 1059.47056

Hitrik, Michael; Sjöstrand, Johannes

Ann. Henri Poincaré 5, No. 1, 1-73 (2004).

The paper is devoted to small non-selfadjoint perturbations of self-adjoint \(h\)-pseudodifferential operators in 2 dimensions. These investigations are motivated by several works of the second named author of the paper [see for example, A. Melin and J. Sjöstrand, Astérisque 284, 181–244 (2003; Zbl 1061.35186)].
In the paper it is considered the case when the classical flow of the unperturbed part is periodic and the strength \(\varepsilon\) of the perturbation is \(>>h\) and bounded from above by \(h^{\delta}\) for some \(\delta>0\). A complete asymptotic description of all eigenvalues in certain rectangles \([-1/C,1/C]+\)
\(i\varepsilon[F_0-1/C,F_0+1/C]\) is obtained.

Reviewer: Farruh Mukhamedov (Tashkent)

MSC:

47G30	Pseudodifferential operators
81Q15	Perturbation theories for operators and differential equations in quantum theory
35P20	Asymptotic distributions of eigenvalues in context of PDEs
35S15	Boundary value problems for PDEs with pseudodifferential operators
81Q10	Selfadjoint operator theory in quantum theory, including spectral analysis
47N50	Applications of operator theory in the physical sciences

Keywords:

\(h\)-pseudodifferential operator; non-selfadjoint perturbation

Citations:

Zbl 1061.35186