Integral transforms and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e23" altimg="si3.svg"><mml:mi mathvariant="script">PT</mml:mi></mml:math>-symmetric Hamiltonians

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