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