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

Abstract

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.

22 pages, LaTeX 2e

References in corpus (1)

• Semi-classical States for Non-self-adjoint Schrodinger Operators

Cited by in corpus (12)

• Pseudospectra of semi-classical (pseudo)differential operators
• From resolvent bounds to semigroup bounds
• Non-elliptic quadratic forms and semiclassical estimates for non-selfadjoint operators
• On the semi-classical analysis of Schrödinger operators with purely imaginary electric potentials in a bounded domain
• Resolvent estimates for non-selfadjoint operators with double characteristics
• The norm of the non-self-adjoint harmonic oscillator semigroup
• Temporal breakdown and Borel resummation in the complex Langevin method
• Resolvent estimates for one-dimensional Schrödinger operators with complex potentials
• Spectral properties of non-self-adjoint operators
• A Bound on the Pseudospectrum of the Harmonic Oscillator with Imaginary Cubic Potential
• Resolvent estimates for non-self-adjoint operators via semi-groups
• The relativistic rotated harmonic oscillator