Invariant current approach to wave propagation in locally symmetric structures

TL;DR

This paper introduces a novel invariant current method for analyzing wave propagation in systems with local inversion and translation symmetries, revealing universal wavefunction structures and enabling efficient computation in complex structures.

Contribution

It develops a new theoretical framework using local symmetry-induced currents to analyze wave functions in locally symmetric potentials, independent of boundary conditions.

Findings

Universal wavefunction structure in locally symmetric potentials

Efficient computation of wave amplitudes in complex systems

Potential for wave localization control in structured media

Abstract

A theory for wave mechanical systems with local inversion and translation symmetries is developed employing the two-dimensional solution space of the stationary Schr\"odinger equation. The local symmetries of the potential are encoded into corresponding local basis vectors in terms of symmetry-induced two-point invariant currents which map the basis amplitudes between symmetry-related points. A universal wavefunction structure in locally symmetric potentials is revealed, independently of the physical boundary conditions, by using special local bases which are adapted to the existing local symmetries. The local symmetry bases enable efficient computation of spatially resolved wave amplitudes in systems with arbitrary combinations of local inversion and translation symmetries. The approach opens the perspective of a flexible analysis and control of wave localization in structurally complex systems.