Spectral Theory of Time Dispersive and Dissipative Systems

TL;DR

This paper develops a spectral theory framework for linear time dispersive and dissipative systems, enabling the extension to conservative systems and analyzing their spectral properties, with applications to dielectrics and damped oscillators.

Contribution

It introduces a mathematically consistent method to extend non-conservative systems to conservative ones and analyzes their spectral characteristics, which was not previously possible.

Findings

Constructive method for conservative extension of dispersive systems

Application to spectral analysis of dielectrics and damped oscillators

Conservative extension of Maxwell equations for lossy dielectrics

Abstract

We study linear time dispersive and dissipative systems. Very often such systems are not conservative and the standard spectral theory can not be applied. We develop a mathematically consistent framework allowing (i) to constructively determine if a given time dispersive system can be extended to a conservative one; (ii) to construct that very conservative system -- which we show is essentially unique. We illustrate the method by applying it to the spectral analysis of time dispersive dielectrics and the damped oscillator with retarded friction. In particular, we obtain a conservative extension of the Maxwell equations which is equivalent to the original Maxwell equations for a dispersive and lossy dielectric medium.