Hermiticity, however, is only a sufficient, but not necessary condition for this to hold. FOr instance, in [1] it was observed that Hamiltonians with potential terms V = x2(ix)a, for a > 0, possess an entirely real and positive spectra. Since then, non-Hermitian Hamiltonian systems have been under intense investigation. Most studies, however, are restricted to Hamiltonians which are not explicitly time depdendent, and deal with eigenvalue problems. In our work, we follow a completely different direction and address explicitly time-dependent Hamiltonians. In particular, we provide time evolution operators, gauge transformations and a perturbative treatment for such Hamiltonians. As a direct application, we compute transition probabilities for a harmonic oscillator perturbed by a cubic non-Hermitian term, in an external linearly polarized laser field.
A strong pre-requiste for our time-dependent formalism to hold is that the Hamiltonian be pseudo-Hermitian. This means that such a Hamiltonian is related to a Hermitian counterpart by a similarity transformation [2]. We have also developed a systematic procedure for finding new equivalence pairs, and have employed it to obtain generalizations of, among others, the Swanson Hamiltonian, anharmonic oscillators preturbed by igpxp and harmonic oscillators pertubed by ixn. These studies are part of an interdisciplinary effort between mathematical and optical physics.
[1] C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
[2] A. Mostafazadeh, J. Math. Phys. 43, 205 (2002).
My collaborators: A. Fring and I. Rotter