Szego condition and propagation of waves
The classical d’Alembert solution of the one-dimensional string equation has the form of two travelling waves: u(x,t) = f(x+t) + g(x-t). If the string is not homogeneous (has a nonconstant density), then the corresponding string equation cannot be solved explicitly. However, the solution, u(x,t), might be asymptotically close to the d’Alembert form at large times t. We prove that this happens if and only if the spectral measure of the string operator belongs to a certain Szego class (has a finite logarithmic integral). Joint work with Sergey Denisov (University of Wisconsin-Madison).