Strings, Waves, Drums: Spectra and Inverse Problems

Abstract

This survey treats a number of interconnected topics related in one way or another to the famous paper of Mark Kac, "Can one hear the shape of a drum?": wave motion, classical and quantum inverse problems, integrable systems, and the relations between spectra and geometry. We sketch the history and some of the principal developments from the vibrating string to quantum inverse problems, the KdV equation and integrable systems, spectral geometry and the index problem.

Measure-geometric Laplacians on the real line

“…Theorem 1.2 shows that the eigenvalues depend on the weights of the Dirac point masses and their positions relative to ν, but that they are independent of the distribution of ν; this condition is different than that given for the Kreȋn-Feller operator ∆ η,Λ , where η is a purely atomic measure, compare with [2].…”

“…The problem in two dimensions remained open until 1992, when Gordon, Webb, and Wolpert [12] constructed a pair of regions in the plane that have different shapes but whose associated Laplacians have identical eigenvalues. Nevertheless, as observed by Weyl [30], Berry [3,4], Lapidus et al [21,22,23,24,25], Beals and Greiner [2] and many others, the spectrum of a Laplacian still tells us a lot about the shape of the underlying geometric structure.…”

“…Arzt [1], Freiberg [7,8,9], Fujita [11], and Kotani and Watanabe [20] have also considered the Kreȋn-Feller operator ∆ η,Λ ≔ ∇ η • ∇ Λ , where η denotes a continuous Borel probability measure and Λ denotes the Lebesgue measure. In the case that η is a purely atomic measure, it has been shown in [2] that the eigenvalues of ∆ η,Λ highly depend on the position and weights of the atoms. Interestingly, the operators ∆ η,Λ and ∆ η,η , in the case that η is continuous, also appear as the infinitesimal generator of Liouville Brownian motion [14] and Liouville quantum gravity [28].…”

Learning Potentials of Quantum Systems using Deep Neural Networks

“…Finding the Hamiltonian operator that generated a given wave-function or rather an effective potential that can generate the wave-function for a quantum system may be related to the famous question made by mathematician Mark Kac [27]: "Can one hear the shape of a drum?" In the case of sound waves, the answer to this question is no for all cases except for the trivial case where the shape of a string is equal to its length [28,29]. In the case of quantum waves, despite the fact that one can get a lot of geometrical and topological information from the spectrum or even its asymptotic behavior, this information is not complete even for quantum systems as simple as the ones defined along a finite interval.…”

Generalised Krein-Feller operators and gap diffusions via transformations of measure spaces

“…The classical Kreȋn-Feller differential operator ∆ ν,Λ , introduced in [8,16], where ν denotes a non-atomic compactly supported Borel probability measure on R and where Λ denotes the one-dimensional Lebesgue measure, has been investigated with respect to its spectral properties first by Fujita [13], Küchler [21], Langer [22] and Kotani and Watanabe [20] and more recently by Arzt [1], Ehnes [6] and Freiberg [9,10,11]. The case when ν is purely atomic has also been studied in [2]; where it was shown that the eigenvalues of ∆ ν,Λ have a dependence not only on the positions of the atoms of ν but also on the weights of the atoms. Returning to the case when ν is a non-atomic, it has been established that ∆ ν,Λ is the infinitesimal generator of a Liouville Brownian motion (also known as gap diffusion, skip-free diffusion, quasi-diffusion or generalised diffusion), see [3,4,7,15,21,22,24,26].…”