Spectral analysis of the discrete Maxwell operator: The limiting absorption principle

TL;DR

This paper investigates the spectral properties of the anisotropic discrete Maxwell operator on a 3D lattice, establishing the limiting absorption principle and analyzing the self-adjointness of conjugate operators at thresholds.

Contribution

It constructs a conjugate operator for the Fourier series of the Maxwell operator and proves the limiting absorption principle for non-zero real values.

Findings

Limiting absorption principle holds for the operator.

Conjugate operator is essentially self-adjoint at certain thresholds.

Spectral analysis advances understanding of discrete Maxwell operators.

Abstract

We are interested by the spectral analysis of the anisotropic discrete Maxwell operator $\hat H^D$ defined on the square lattice $\rm Z\!\!\! Z^3$. In aim to prove that the limiting absorption principle holds we construct a conjugate operator to the Fourier series of $\hat H^D$ at any not-zero real value. In addition we show that at some particular thresholds the conjugate operator is essentially self-adjoint.