Spectral Properties of an Acoustic Boundary Condition
A boundary condition is studied for the wave equation occurring in theoretical acoustics. The initial value problem in a bounded domain is solved by semigroup methods in a Hilbert space of data with finite energy. A description of the spectrum of the semigroup generator A is then obtained. Unlike the generators associated with the usual boundary conditions, which have compact resolvent and spectrum consisting of discrete eigenvalues, A always has essential spectrum. Moreover, if the parameters occurring in the boundary condition are constant, there are sequences of eigenvalues converging to the essential spectrum.