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

A family of exactly solvable quantum square wells with discrete coordinates and with certain non-stationary Hermiticity-violating Robin boundary conditions is proposed and studied. Manifest non-Hermiticity of the model in conventional Hilbert space <?CDATA H _friendly?> H friendly is required to coexist with the unitarity of system in another, ad hoc Hilbert space <?CDATA H _physical?> H physical . Thus, quantum mechanics in its non-Hermitian interaction picture (NIP) representation is to be used. We must construct the time-dependent states (say, ψ ( t )) as well as the time-dependent observables (say, Λ( t )). Their evolution in time is generated by the operators denoted, here, by the respective symbols G ( t ) (a Schrödinger-equation generator) and Σ( t ) (a Heisenberg-equation generator, a.k.a. quantum Coriolis force). The unitarity of evolution in <?CDATA H _physical?> H physical is then guaranteed by the reality of spectrum of the energy observable alias Hamiltonian H ( t ) = G ( t ) + Σ( t ). The applicability of these ideas is illustrated via an N by N matrix model. At N = 2, closed formulae are presented not only for the measurable instantaneous energy spectrum but also for all of the eligible time-dependent physical inner-product metrics Θ ( N =2) ( t ), for the related Dyson maps Ω ( N =2) ( t ), for the Coriolis force Σ ( N =2) ( t ) as well as, in the very ultimate step of the construction, for the truly nontrivial Schrödinger-equation generator G ( N =2) ( t ).