Mending the broken PT-Symmetry in non-Hermitian quantum mechanics

It is known that non-Hermitian quantum systems with PT-symmetry exhibit real eigenvalues and have unitary time evolutions. When this symmetry is spontaneously broken, the eigenvalues become complex and we would ordinarily dismiss the system as meaningless. However, we demonstrate that non-Hermitian Hamiltonians with broken PT-symmetry can be made meaningful when introducing some explicit time dependence into their parameters.

We show that explicitly time dependent non-Hermitian Hamiltonians lose their dual nature, that is simultaneously satisfying the Schrodinger equation and acting as the energy operator. Instead the Hamiltonian becomes unobservable and we must define a new energy operator. We use three separate methods to solve such systems; namely the Dyson equation, the quasi-Hermiticity equation and the Lewis Riesenfeld Invariant method.

References
[1] A. Fring and T. Frith. Mending the broken PT -regime via an explicit time-dependent Dyson map. Phys. Lett. A 381(29), 2318 { 2323 (2017).
[2] A. Fring and T. Frith. Solvable two-dimensional time-dependent non-Hermitian quantum systems with infinite dimensional Hilbert space in the broken PT -regime. J. Phys. A: Math. Theor. 51(26), 265301 (2018).
[3] A. Fring and T. Frith. Quasi-exactly solvable quantum systems with explicitly time-dependent Hamiltonians. Phys. Lett. A 383(2-3), 158{163 (2019).
[4] A. Fring and T. Frith. Time-dependent metric for the two dimensional, non-Hermitian coupled oscillator. arXiv preprint arXiv:1812.02862 (2018).