PT-symmetric quantum graphs

TL;DR

This paper explores PT-symmetric quantum graphs, deriving boundary conditions that ensure real eigenvalues and positive norms, and discusses their potential realization in optical waveguides.

Contribution

It introduces general PT-symmetric boundary conditions for quantum graphs and provides explicit forms, expanding the understanding of PT-symmetry in branched quantum systems.

Findings

Derived general PT-symmetric boundary conditions for quantum graphs.

Presented explicit boundary conditions consistent with PT-symmetry.

Discussed experimental realization in optical waveguides.

Abstract

We consider branched quantum wires, whose connection rules provide PT-symmetry for the Schrodinger equation on graph. For such PT-symmetric quantum graph we derive general boundary conditions which keep the Hamiltonian as PT-symmetric with real eigenvalues and positively defined norm. Explicit boundary conditions which are consistent with the general PT-symmetric boundary conditions are presented. Secular equations for finding the eigenvalues of the quantum graph are derived. Breaking of the Kirchhoff rule at the branching point is shown. Experimental realization of PT-symmetric quantum graphs on branched optical waveguides is discussed.