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

Integral transforms can be used as a tool to simplify the computations of differential equations. In this work, we systematically study integral transforms in the context of $\mathcal{PT}$-symmetric Hamiltonians . First, we obtained a closed analytical formula for the exponential Fourier transformed $\mathcal{PT}$-symmetric Hamiltonians. Using Segal-Bargmann transform, we investigate the effect of the Fourier transform on the eigenfunctions of the original Hamiltonian. Moreover, 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 complex Hilbert space. Specifying to non-Hermitian $XX$ spin chain, we prove by numerically solving the quantum master equation its ability to flip from dynamical to static system by running the coupling constant from weak to strong. Finally, we solve the Swanson Hamiltonian and discuss its behavior under integral transforms.