Spectral instability of some non-selfadjoint anharmonic oscillators

TL;DR

This paper investigates the spectral instability of certain non-selfadjoint differential operators, specifically the complex Airy operator and anharmonic oscillators, by analyzing the growth of spectral projection norms for large eigenvalues.

Contribution

It provides asymptotic expansions for the norms of spectral projections of these operators, extending previous results by Davies and Davies-Kuijlaars.

Findings

Spectral projections grow rapidly for large eigenvalues.

Asymptotic formulas describe the growth rate of spectral norms.

Results highlight the instability of spectra in non-selfadjoint operators.

Abstract

The purpose of this Note is to highlight the spectral instability of some non-selfadjoint differential operators, by studying the growth rate of the norms of the spectral projections $\Pi_n$ associated with their eigenvalues. More precisely, we are concerned with the complex Airy operator and even anharmonic oscillator. We get asymptotic expansions for the norm of the spectral projections associated with the large eigenvalues, extending the results of Davies and Davies-Kuijlaars.