Zig-zag-matrix algebras and solvable quasi-Hermitian quantum models

Miloslav Znojil

doi:10.1088/1751-8121/ace8d5

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arXiv · 2307.03439

Zig-zag-matrix algebras and solvable quasi-Hermitian quantum models

Miloslav Znojil

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Abstract

It is well known that the unitary evolution of a closed $M-$level quantum system can be generated by a non-Hermitian Hamiltonian $H$ with real spectrum. Its Hermiticity can be restored via an amended inner-product metric $\Theta$. In Hermitian cases the evaluation of the spectrum (i.e., of the bound-state energies) is usually achieved by the diagonalization of the Hamiltonian. In the non-Hermitian (or, more precisely, in the $\Theta-$quasi-Hermitian) quantum mechanics we conjecture that the role of the diagonalized-matrix solution of the quantum bound-state problem could be transferred to a maximally sparse ``zig-zag-matrix'' representation of the Hamiltonians.

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Miloslav Znojil. 2023-07-07. Zig-zag-matrix algebras and solvable quasi-Hermitian quantum models. https://doi.org/10.1088/1751-8121/ace8d5

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