The Damped String Problem Revisited

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.Comment: 39 page

Eliciting Harmonics on Strings

International audienceOne may produce the $q$th harmonic of a string of length $\pi$ by applying the 'correct touch' at the node $\pi/q$ during a simultaneous pluck or bow. This notion was made precise by a model of Bamberger, Rauch and Taylor. Their 'touch' is a damper of magnitude $b$ concentrated at $\pi/q$. The 'correct touch' is that $b$ for which the modes, that do not vanish at $\pi/q$, are maximally damped. We here examine the associated spectral problem. We find the spectrum to be periodic and determined by a polynomial of degree $q-1$. We establish lower and upper bounds on the spectral abscissa and show that the set of associated root vectors constitutes a Riesz basis and so identify 'correct touch' with the $b$ that minimizes the spectral abscissa