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

M. W. AlMasri, M. R. B. Wahiddin

Published 2022-07-06Version 1

The exponential Fourier transform of a given non-Hermitian $\mathcal{PT}$-symmetric potential in the position space is Hermitian . We prove this proposition for any $\mathcal{PT}$-symmetric non-Hermitian Hamiltonians. The hermiticity of the Fourier transform of non-Hermitian Hamiltonian operator can be used as a condition for the reality of energy spectra in the unbroken $\mathcal{PT}$-symmetric regime . In the broken $\mathcal{PT}$-symmetric regime, pairs of complex eigenvalues may appear for potentials written in the position space. However, these complex pairs disappear in the momentum space and we are left only with real eigenvalues. Finally, we test our proposition in the case of Swanson Hamiltonian.