Transition probabilities for non self-adjoint Hamiltonians in infinite dimensional Hilbert spaces

TL;DR

This paper extends the study of transition probabilities for non self-adjoint Hamiltonians from finite to infinite dimensional Hilbert spaces, analyzing models like the harmonic oscillator and Landau levels to uncover new physical insights.

Contribution

It introduces the analysis of non self-adjoint Hamiltonians in infinite dimensional spaces, expanding previous finite-dimensional results to more realistic quantum systems.

Findings

New features in transition probabilities for infinite-dimensional systems

Application to models like the extended harmonic oscillator and Landau levels

Deeper understanding of non self-adjoint quantum dynamics

Abstract

In a recent paper we have introduced several possible inequivalent descriptions of the dynamics and of the transition probabilities of a quantum system when its Hamiltonian is not self-adjoint. Our analysis was carried out in finite dimensional Hilbert spaces. This is useful, but quite restrictive since many physically relevant quantum systems live in infinite dimensional Hilbert spaces. In this paper we consider this situation, and we discuss some applications to well known models, introduced in the literature in recent years: the extended harmonic oscillator, the Swanson model and a generalized version of the Landau levels Hamiltonian. Not surprisingly we will find new interesting features not previously found in finite dimensional Hilbert spaces, useful for a deeper comprehension of this kind of physical systems.