The complete menu of eligible metrics for a family of toy Hamiltonians $H\ne H^{†}$ with real spectra

Abstract

An elementary set of non-Hermitian $N$ by $N$ matrices $H^{(N)}(\lambda )\ne [H^{(N)}(\lambda ){]}^{†}$ with real spectra is considered, assuming that each of these matrices represents a selfadjoint quantum Hamiltonian in an {\it ad hoc} Hilbert space of states ${\mathcal{H}}^{(physical)}$. The problem of an explicit specification of all of these spaces (i.e., in essence, of all of the eligible {\it ad hoc} inner products and metric operators $\Theta$) is addressed. The problem is shown exactly solvable and, for every size $N=2,4,...$ and parameter $\lambda \in (-1,1)$ in matrix $H^{(N)}(\lambda )$, the complete $N-$parametric set of metrics ${\Theta }_{{\alpha }_1,{\alpha }_2,...,{\alpha }_N}^{(N)}(\lambda )$ is recurrently defined by closed formula.

28 pp., 2 figures, thoroughly rewritten

References in corpus (1)

• Interface between Hermitian and non-Hermitian Hamiltonians in a model calculation