Can One Hear the Composition of a Drum?

Abstract

This note is related to the famous question raised by Mark Kac and concerning the determination of the shape of the drum by the eigenvalues of its governing equation. Here, we allow the drum to be composed by several different types of membranes and we consider the problem of hearing the composition of the drum, starting from the eigenvalues of numerical approximations of the related equation. Some key tools, taken from asymptotic linear algebra, are reported and extended, and allow somehow to answer to the question in the positive. (2000). Primary 65F10; Secondary 15A18. Mathematics Subject Classification

The asymptotic properties of the spectrum of nonsymmetrically perturbed Jacobi matrix sequences

“…A special part of them is the GLT theory (see [16,17] and references therein) which allows to treat the case of variable coefficients under very mild restrictions on the regularity of the coefficients (e.g., numerical approximations of variable coefficient PDEs [16] and systems of PDEs [17], Jacobi sequences with asymptotically varying periodic [5] and non-periodic [11] coefficients, etc.). The interesting fact is that the tools explicitly developed here are applicable verbatim to these cases as well (see [9] for an example), by allowing to deal with non-Hermitian perturbations under the same mild trace conditions.…”

On the Asymptotic Spectrum of Finite Element Matrix Sequences

“…Finally, recently the above results have been extended to non-Hermitian matrices A n occurring, e.g., in the finite difference discretization of PDEs containing lower order difference operators: it has been shown in [16,18] that, provided that the spectral norm of A n is uniformly bounded in n and that the trace norm of S n = (A n −A * n )/(2i), the skew-Hermitian part of A n , grows at most as o(n), then the sequence (A n ) has the same asymptotic spectrum as the sequence ((A n + A * n )/2) obtained from the Hermitian part of A n . This result also implies [18,19] that (9) remains true for more general domains Ω, even if one uses different approximation schemes for the boundary conditions.…”

Stability of the notion of approximating class of sequences and applications

“…In fact T −1 n (2 + 2 cos(s))T n ((2 + 2 cos(s))h(s)) can be written as T n (h) plus a correction term E n whose rank is at most equal to two. If E n (or some symmetrized version) has a trace norm infinitesimal with respect to n, then the result could be proven by combining Theorem 4.1 and the tools developed in [36] (see also [10,16]). …”

“…(6) and (7)], can be proven as a consequence of the same result. Finally, we observe that the tools considered in this paper can be used for providing an alternative proof of the stability under inversion of the large class of the GLT sequences, whose importance depends also on the fact that the GLT class includes any approximation of PDEs (see [29,16,31,2]) by local methods such as finite differences, finite elements, and finite volumes. The precise investigation in these directions is planned to be pursued in future research.…”

“…Golinskii and Serra-Capizzano[10], Holmgren et al[16]). Let {A n } n be a sequence of square matrices of size d n , let H n = Re(A n ), S n = Im(A n ), and let us denote by X trace the trace norm of a matrix X (i.e., the sum of its singular values or Schatten p norm with p = 1).1.…”