Discrete-coordinate crypto-Hermitian quantum system controlled by time-dependent Robin boundary conditions

Abstract: Non-stationary version of unitary quantum mechanics formulated in non-Hermitian (or, more precisely, in hiddenly Hermitian) interaction-picture representation is illustrated via an elementary $N$ by $N$ matrix Hamiltonian $H(t)$ mimicking a 1D-box system with physics controlled by time-dependent boundary conditions. The model is presented as analytically solvable at $N=2$. Expressis verbis, this means that for both of the underlying Heisenbergian and Schr\"{o}dingerian evolution equations the generators (i.e., in our notation, the respective operators $\Sigma(t)$ and $G(t)$) become available in closed form. Our key message is that contrary to the conventional beliefs and in spite of the unitarity of the evolution of the system, neither its "Heisenbergian Hamiltonian" $\Sigma(t)$ nor its "Schr\"{o}dingerian Hamiltonian" $G(t)$ possesses a real spectrum or the conjugate pairs of complex eigenvalues. This means that neither one of these "Hamiltonians" can be pseudo-Hermitian alias PT-symmetric.