Hamiltonian and Linear-Space Structure for Damped Oscillators: I. General Theory

TL;DR

This paper develops a Hamiltonian and linear-space framework for damped oscillators, enabling the use of self-adjointness concepts to analyze their properties similarly to conservative systems.

Contribution

It introduces a bilinear map structure for damped oscillators' phase space, allowing eigenvector orthogonality, sum rules, and perturbation theory to be applied.

Findings

Eigenvectors are orthogonal under the bilinear map.

Sum rules analogous to conservative systems are established.

Perturbation theory applies to complex eigenvalues.

Abstract

The phase space of $N$ damped linear oscillators is endowed with a bilinear map under which the evolution operator is symmetric. This analog of self-adjointness allows properties familiar from conservative systems to be recovered, e.g., eigenvectors are "orthogonal" under the bilinear map and obey sum rules, initial-value problems are readily solved and perturbation theory applies to the_complex_ eigenvalues. These concepts are conveniently represented in a biorthogonal basis.