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

In quantum mechanics of unitary systems using non-Hermitian (or, more precisely, Theta-quasi-Hermitian) Hamiltonians H such that H(SIC) Theta = Theta H, the exactly solvable M-level bound-state models with arbitrary M <=infinity 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 Theta) was achieved due to an extremely sparse (viz., just (2M-1)- parametric) but still nontrivial 'zig-zag-matrix' choice of the form of H.