Spectral analysis of a selected non self-adjoint Hamiltonian in an infinite dimensional Hilbert space

TL;DR

This paper explores the spectral properties of a specific non self-adjoint Hamiltonian in an infinite-dimensional Hilbert space, using the equation of motion method and pseudo-bosons to analyze eigenvalues and eigenvectors.

Contribution

It introduces a spectral analysis approach for a class of non self-adjoint Hamiltonians using pseudo-bosons, providing explicit eigenstructure in infinite dimensions.

Findings

Eigenvalues are real for the studied Hamiltonian.

Eigenvectors can be explicitly constructed using pseudo-bosons.

The method facilitates diagonalization of non self-adjoint operators.

Abstract

The so-called equation of motion method is useful to obtain the explicit form of the eigenvectors and eigenvalues of certain non self-adjoint bosonic Hamiltonians with real eigenvalues. These operators can be diagonalized when they are expressed in terms of pseudo-bosons, which do not behave as ordinary bosons under the adjoint transformation, but obey the Weil-Heisenberg commutation relations.