Anisotropy and Asymptotic Degeneracy of the Physical-Hilbert-Space Inner-Product Metrics in an Exactly Solvable Unitary Quantum Model

0584994 - ÚJF 2025 RIV CH eng J - Článek v odborném periodiku
Znojil, Miloslav
Anisotropy and Asymptotic Degeneracy of the Physical-Hilbert-Space Inner-Product Metrics in an Exactly Solvable Unitary Quantum Model.
Symmetry-Basel. Roč. 16, č. 3 (2024), č. článku 353. E-ISSN 2073-8994
Institucionální podpora: RVO:61389005
Klíčová slova: unitary quantum solvable toy model * Hilbert-space anisotropy * eigenvalues of metric * exceptional-point collapse
Obor OECD: Atomic, molecular and chemical physics (physics of atoms and molecules including collision, interaction with radiation, magnetic resonances, Mössbauer effect)
Impakt faktor: 2.7, rok: 2022
Způsob publikování: Open access
https://doi.org/10.3390/sym16030353

A unitary-evolution process leading to an ultimate collapse and to a complete loss of observability alias quantum phase transition is studied. A specific solvable N-state model is considered, characterized by a non-stationary non-Hermitian Hamiltonian. Our analysis uses quantum mechanics formulated in Schrodinger picture in which, in principle, only the knowledge of a complete set of observables (i.e., operators Lambda j) enables one to guarantee the uniqueness of the related physical Hilbert space (i.e., of its inner-product metric Theta). Nevertheless, for the sake of simplicity, we only assume the knowledge of just a single input observable (viz., of the energy-representing Hamiltonian H equivalent to Lambda 1). Then, out of all of the eligible and Hamiltonian-dependent 'Hermitizing' inner-product metrics Theta=Theta(H), we pick up just the simplest possible candidate. Naturally, this slightly restricts the scope of the theory, but in our present model, such a restriction is more than compensated for by the possibility of an alternative, phenomenologically better motivated constraint by which the time-dependence of the metric is required to be smooth. This opens a new model-building freedom which, in fact, enables us to force the system to reach the collapse, i.e., a genuine quantum catastrophe as a result of the mere conventional, strictly unitary evolution.
Trvalý link: https://hdl.handle.net/11104/0352773