The complete menu of eligible metrics for a family of toy Hamiltonians $H \neq H^\dagger$ with real spectra

Abstract

An elementary set of non-Hermitian $N$ by $N$ matrices $H^{(N)}(λ) \neq [ H^{(N)}(λ)]^\dagger$ 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 ${\cal H}^{(physical)}$. 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 $Θ$) is addressed. The problem is shown exactly solvable and, for every size $N=2,4,...$ and parameter $λ\in (-1,1)$ in matrix $H^{(N)}(λ)$, the complete $N-$parametric set of metrics $Θ^{(N)}_{α_1,α_2,...,α_N}(λ)$ is recurrently defined by closed formula.