  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.
