Non self-adjoint Hamiltonians with complex eigenvalues

TL;DR

This paper explores the properties of non self-adjoint, diagonalizable Hamiltonians with complex eigenvalues in finite-dimensional quantum systems, focusing on their mathematical structure and physical implications in PT-symmetric contexts.

Contribution

It introduces a framework for analyzing such Hamiltonians using antilinear operators and provides a detailed study of transition probabilities in different regimes.

Findings

Intertwining relations can be established with antilinear operators.

Transition probabilities differ significantly between broken and unbroken regimes.

The mathematical structure extends PT-symmetric quantum mechanics to non-Hermitian Hamiltonians.

Abstract

Motivated by what one observes dealing with PT-symmetric quantum mechanics, we discuss what happens if a physical system is driven by a diagonalizable Hamiltonian with not all real eigenvalues. In particular, we consider the functional structure related to systems living in finite-dimensional Hilbert spaces, and we show that certain intertwining relations can be deduced also in this case if we introduce suitable antilinear operators. We also analyze a simple model, computing the transition probabilities in the broken and in the unbroken regime.