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

Abstract

Abstract In quantum mechanics of unitary systems using non-Hermitian (or, more precisely, Θ-quasi-Hermitian) Hamiltonians H such that H † Θ = Θ H , the exactly solvable M-level bound-state models with arbitrary M ⩽ ∞ are rare. A new class of such models is proposed here, therefore. Its exact algebraic solvability (involving not only the closed formulae for wave functions but also the explicit description of all of the eligible metrics Θ) was achieved due to an extremely sparse (viz., just ( 2 M − 1 ) -parametric) but still nontrivial ‘zig-zag-matrix’ choice of the form of H.