Integral Transforms and $\mathcal{PT}$-symmetric Hamiltonians

TL;DR

This paper explores the role of integral transforms, especially the Fourier and Segal-Bargmann transforms, in analyzing $\\mathcal{PT}$-symmetric Hamiltonians, revealing their effects on eigenfunctions and applications to non-Hermitian systems.

Contribution

It provides a systematic study of integral transforms for $\\mathcal{PT}$-symmetric Hamiltonians, including analytical formulas and applications to holomorphic representations.

Findings

Derived a closed-form for the exponential Fourier transform of $\\mathcal{PT}$-symmetric Hamiltonians.

Analyzed the impact of Fourier transform on eigenfunctions via the Segal-Bargmann transform.

Discussed integral transforms' role in understanding the Swanson Hamiltonian.

Abstract

Motivated by the fact that twice the Fourier transform plays the role of parity operator. We systematically study integral transforms in the case of $\mathcal{PT}$-symmetric Hamiltonian. First, we obtain a closed analytical formula for the exponential Fourier transform of a general $\mathcal{PT}$-symmetric Hamiltonian. Using the Segal-Bargmann transform, we investigate the effect of the Fourier transform on the eigenfunctions of the original Hamiltonian. As an immediate application, we comment on the holomorphic representation of non-Hermitian spin chains, in which the Hamiltonian operator is written in terms of analytical phase-space coordinates and their partial derivatives in the Bargmann space rather than matrices in the vector Hilbert space. Finally, we discuss the effect of integral transforms in the study of the Swanson Hamiltonian.