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Acta Polytechnica

Swanson Hamiltonian revisited through the complex scaling method

M. Reboiro, Romina Ramírez, Viviano Fernández

Published Feb 28, 2022• 4 citations
Publisher

Abstract

In this work, we study the non-hermitian PT-symmetry Swanson Hamiltonian in the framework of the Complex Scaling Method. We show that by applying this method we can work with eigenfunctions that are square-integrable both in the PT and in the non-PT symmetry phase.
DOI
10.14311/ap.2022.62.0157

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In the context of two particularly interesting non-Hermitian models in quantum mechanics we explore the relationship between the original Hamiltonian H and its Hermitian counterpart h, obtained from H by a similarity transformation, as pointed out by Mostafazadeh. In the first model, due to Swanson, h turns out to be just a scaled harmonic oscillator, which explains the form of its spectrum. However, the transformation is not unique, which also means that the observables of the original theory are not uniquely determined by H alone. The second model we consider is the original PT-invariant Hamiltonian, with potential V = igx 3. In this case the corresponding h, which we are only able to construct in perturbation theory, corresponds to a complicated velocity-dependent potential. We again explore the relationship between the canonical variables x and p and the observables X and P. 1