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On the spectrum of certain non-commutative harmonic oscillators and semiclassical analysis. (English) Zbl 1168.47038

The author studies matrix differential operators of the form \(A( -\frac{\partial_x^2}2+\frac{x^2}2) +J( x\partial_x+\frac12)\) where \(A=\left(\begin{smallmatrix} \alpha & 0\\ 0 & \beta \end{smallmatrix}\right)\), \(J= \left(\begin{smallmatrix} 0 & -1\\ 1 & 0 \end{smallmatrix}\right)\), \(\alpha ,\beta \in \mathbb R_+\), \(\alpha \neq \beta\), \(\alpha \beta >1\). Using semiclassical techniques, he obtains detailed information on the localization and multiplicity of the spectrum.

MSC:

47E05 General theory of ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
35S05 Pseudodifferential operators as generalizations of partial differential operators
47G30 Pseudodifferential operators
47N50 Applications of operator theory in the physical sciences
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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