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

TL;DR

This paper explicitly constructs all possible inner products and metrics for a family of non-Hermitian matrices with real spectra, demonstrating the exact solvability of the problem for various matrix sizes and parameters.

Contribution

It provides a complete, explicit set of metrics for non-Hermitian Hamiltonians with real spectra, advancing the understanding of their physical Hilbert spaces.

Findings

Explicit formulas for metrics are derived for all matrix sizes and parameters.

The problem of defining physical inner products for these Hamiltonians is shown to be exactly solvable.

A recursive method to obtain the metrics is established.

Abstract

An elementary set of non-Hermitian $N$ by $N$ matrices $H^{(N)}(\lambda) \neq [ H^{(N)}(\lambda)]^\dagger$ 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 $\Theta$) 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 $\Theta^{(N)}_{\alpha_1,\alpha_2,...,\alpha_N}(\lambda)$ is recurrently defined by closed formula.