One can hear whether a drum has finite area
Colin Clark, Denton Hewgill · Proceedings of the American Mathematical Society · 1967
where Np(X)= )j,)jfjp(x) (4j(x)) 2dx, in which p(x) is an arbitrary nonnegative function in L1 (s), and { Xi }, {I 4} are the eigenvalues and eigenfunctions, respectively, of the Laplacian in U. (This formula has also been derived, under other conditions than in [1], by Hewgill [2].) The recent paper by Kac [3] prompts the question: is there some distinction asymptotically between the eigenvalues for a region of finite volume (e.g. a bounded region) and those of a quasi-bounded region of infinite volume?