Quantum inner-product metrics via recurrent solution of Dieudonne equation

TL;DR

This paper explores how to construct inner-product metrics for non-Hermitian Hamiltonians with real spectra using recurrent solutions to the Dieudonne equation, enabling non-numerical analysis of complex lattice Hamiltonians.

Contribution

It demonstrates that all such metrics can be obtained through recurrent solutions of the Dieudonne equation, including for complex lattice Hamiltonians, providing a new analytical approach.

Findings

Metrics are obtainable as recurrent solutions of the Dieudonne equation.

The approach is applicable to N-site lattice Hamiltonians.

Non-numerical solutions are feasible for complex Hamiltonians.

Abstract

A given Hamiltonian matrix H with real spectrum is assumed tridiagonal and non-Hermitian. Its possible Hermitizations via an amended, ad hoc inner-product metric are studied. Under certain reasonable assumptions, all of these metrics are shown obtainable as recurrent solutions of the hidden Hermiticity constraint called Dieudonne equation. In this framework even the two-parametric Jacobi-polynomial real- and asymmetric-matrix N-site lattice Hamiltonian is found tractable non-numerically at all N.