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

0584744 - ÚJF 2025 RIV GB eng J - Journal Article
Znojil, Miloslav
Discrete-coordinate crypto-Hermitian quantum system controlled by time-dependent Robin boundary conditions.
Physica Scripta. Roč. 99, č. 3 (2024), č. článku 035250. ISSN 0031-8949. E-ISSN 1402-4896
Institutional support: RVO:61389005
Keywords : quantum theory of unitary systems * non-Hermitian interaction representation * non-stationary physical inner products * solvable discrete square well
OECD category: Atomic, molecular and chemical physics (physics of atoms and molecules including collision, interaction with radiation, magnetic resonances, Mössbauer effect)
Impact factor: 2.9, year: 2022
Method of publishing: Open access
https://doi.org/10.1088/1402-4896/ad298b

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 Hfriendly is required to coexist with the unitarity of system in another, ad hoc Hilbert space Hphysical . Thus, quantum mechanics in its non-Hermitian interaction picture (NIP) representation is to be used. We must construct the time-dependent states (say, psi(t)) as well as the time-dependent observables (say, Lambda(t)). Their evolution in time is generated by the operators denoted, here, by the respective symbols G(t) (a Schrodinger-equation generator) and sigma(t) (a Heisenberg-equation generator, a.k.a. quantum Coriolis force). The unitarity of evolution in Hphysical is then guaranteed by the reality of spectrum of the energy observable alias Hamiltonian H(t) = G(t) + sigma(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 Theta(N=2)(t), for the related Dyson maps omega(N=2)(t), for the Coriolis force sigma(N=2)(t) as well as, in the very ultimate step of the construction, for the truly nontrivial Schrodinger-equation generator G (N=2)(t).
Permanent Link: https://hdl.handle.net/11104/0352589