# Non-Hermitian quantum mechanics of bosonic operators

## TL;DR

This paper investigates the spectral properties of a non-Hermitian unbounded operator in quantum physics, demonstrating diagonalization and completeness of eigenfunctions, while highlighting the challenges in forming Riesz bases.

## Contribution

It provides a detailed spectral analysis of a non-Hermitian operator, showing diagonalization and completeness, and discusses the limitations in basis properties within quantum mechanics.

## Key findings

- Eigenfunctions form complete systems but are not Riesz bases.
- The operator and its adjoint can be diagonalized using constructed operators.
- Spectral instabilities distinguish these operators from Hermitian Hamiltonians.

## Abstract

The spectral analysis of a non-Hermitian unbounded operator appearing in quantum physics is our main concern. The properties of such an operator are essentially different from those of Hermitian Hamiltonians, namely due to spectral instabilities. We demonstrate that the considered operator and its adjoint can be diagonalized when expressed in terms of certain conveniently constructed operators. We show that their eigenfunctions constitute complete systems, but do not form Riesz bases. Attempts to overcome this difficulty in the quantum mechanical set up are pointed out.