Hamiltonian Extension and Eigenfunctions for a Time Dispersive Dissipative String

We carry out a detailed analysis of a time dispersive and dissipative (TDD) string, using our recently developed theories of conservative and Hamiltonian extensions of TDD systems. This analysis of the TDD string includes, in particular: (i) an explicit construction of its conservative Hamiltonian extension, consisting of the original string coupled to “hidden strings”; (ii) explicit formulas for energy and momentum densities in the extended system, providing a transparent physical picture accounting precisely for the dispersion and dissipation; (iii) the eigenmodes for the extended string system, which provide an eigenmode expansion for solutions to the TDD wave equation governing the solution to the TDD string. The obtained results provide a solid basis for the rigorous treatment of the long standing problem of scattering by a TDD scatterer, illustrated here by the computation of scattering states for a string with dissipation restricted to a half line.