Non-Hermitian Quantum Mechanics of Bosonic Operators

Abstract

The spectral analysis of a family of non-Hermitian operators appearing in quantum physics is our main concern. The properties of such operators are essentially different from those of Hermitian Hamiltonians, namely due to spectral instabilities. We demonstrate that the considered operators and their adjoints 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.