Integral transforms and PT-symmetric Hamiltonians

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