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Journal of Operator Theory

Volume 47, Issue 2, Spring 2002  pp. 413-429.

Non-self-adjoint harmonic oscillator, compact semigroups and pseudospectra

Authors:  Lyonell S. Boulton
Author institution: Departimento de Matematicas, Universidad Simon Bolivar, Apartado 89000, Caracas 1080-A, Venezuela

Summary:  We provide new information concerning the pseudospectra of the complex harmonic oscillator. Our analysis illustrates two different techniques for getting resolvent norm estimates. The first uses the JWKB method and extends for this particular potential some results obtained recently by E.B.~Davies. The second relies on the fact that the bounded holomorphic semigroup generated by the complex harmonic oscillator is of Hilbert-Schmidt type in a maximal angular region. In order to show this last property, we deduce a non-self-adjoint version of the classical Mehler's formula.

Keywords:  Complex harmonic oscillator, non-self-adjoint, resolvent norm estimates, bounded holomorphic semigroups, pseudospectrum, JWKB method, Mehler's formula

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