Cryptohermitian Hamiltonians on graphs. II. Hermitizations

TL;DR

This paper investigates non-hermitian quantum graphs with real spectra by discretizing them and constructing metrics that render the Hamiltonians Hermitian in different mathematical frameworks, enhancing their physical interpretability.

Contribution

It introduces methods to Hermitize non-hermitian quantum graph Hamiltonians using pseudometrics and metrics, facilitating their analysis within quantum mechanics.

Findings

Discretization of non-hermitian quantum graphs with real spectra.

Construction of pseudometrics and metrics to Hermitize Hamiltonians.

Hamiltonians become Hermitian in auxiliary or physical Hilbert spaces.

Abstract

Non-hermitian quantum graphs possessing real (i.e., in principle, observable) spectra are studied via their discretization. The discretized Hamiltonians are assigned, constructively, an elementary pseudometric and/or a more complicated metric. Both these constructions make the Hamiltonian Hermitian, respectively, in an auxiliary (Krein or Pontryagin) vector space or in a less friendly (but more useful) Hilbert space of quantum mechanics.