Pseudo-Spectra of The Complex Harmonic Oscillator

This paper is devoted to study the pseudo-spectra of the complex harmonic oscillator. The pseudo-spectra of an operator are subsets in the complex plane where the resolvent is large in norm. The non-self-adjoint harmonic oscillator has recently been explored by E.B. Davies, L.S. Boulton and N. Trefethen. In a paper of L.S. Boulton, he proves that the resolvent norm of the complex harmonic oscillator tends to infinity along the curves of the form: z_η = bη + cη^p where independent of η 0 and c is a complex number such that Re(c) and I m(c) < 0. This permits to precise the shape of the pseudo-spectra and to understand the stability of the spectrum under small perturbations. In the present paper, we study the behavior of the resolvent norm along z_η for and we verify the spectral instability using some numerical computations of pseudo-spectra.