Riesz property in the case of multiple eigenvalues

TL;DR

This paper investigates the Riesz property of spectral projections for non-symmetric perturbations of self-adjoint operators, including cases with multiple and infinite eigenvalues, and establishes conditions for specific operators with complex potentials.

Contribution

It extends the understanding of the Riesz property to operators with multiple eigenvalues and complex perturbations, including classical quantum operators.

Findings

Established the Riesz property for perturbations of the multi-dimensional harmonic oscillator.

Proved the Riesz property for the Landau Hamiltonian under certain complex potentials.

Extended results to the Laplace-Beltrami operator on a sphere with complex $L^r$-potentials.

Abstract

We analyze spectra and the Riesz property of spectral projections of non-symmetric perturbations of self-adjoint operators with eigenvalues having arbitrary multiplicities, including infinite ones. In particular, we establish the Riesz property for perturbations of the multi-dimensional harmonic oscillator, Landau Hamiltonian and Laplace-Beltrami operator on a sphere by complex-valued $L^r$-potentials if $d/2 < r < \infty$.