The Damped String Problem Revisited

TL;DR

This paper revisits the damped string problem, deriving new trace formulas and basis results using operator theory, under less restrictive smoothness assumptions on coefficients.

Contribution

It introduces novel trace formulas and completeness results for the damped string equation, employing operator theoretic methods and relaxing smoothness conditions.

Findings

Derived an infinite sequence of trace formulas

Established completeness and Riesz basis results

Applied operator theoretic methods to less smooth coefficients

Abstract

We revisit the damped string equation on a compact interval with a variety of boundary conditions and derive an infinite sequence of trace formulas associated with it, employing methods familiar from supersymmetric quantum mechanics. We also derive completeness and Riesz basis results (with parentheses) for the associated root functions under less smoothness assumptions on the coefficients than usual, using operator theoretic methods (rather than detailed eigenvalue and root function asymptotics) only.