Short-time asymptotics of the trace of the wave operator for a general annular drum in R2 with Robin boundary conditions

Short-time asymptotics of the trace of the wave operator <(<mu>)over cap>(t)= [GRAPHICS] exp(-it mu (1/2)(j)), where {mu}(j=1)(infinity) are the eigenvalues of the negative Laplacian in R-2, is Studied for a variety of domains, where -infinity < t < infinity and i = root -1. The dependance of <(<mu>)over cap>(t) on the connectivity of bounded drums and the Robin boundary conditions are analyzed. Particular attention is given to a general annular drum in R-2 together with the Robin boundary conditions on its boundaries.