Related papers: Coincidence of length spectra does not imply isosp…
Isospectral domains are non-isometric regions of space for which the spectra of the Laplace-Beltrami operator coincide. In the two-dimensional Euclidean space, instances of such domains have been given. It has been proved for these examples…
Isospectrality of planar domains which are obtained by successive unfolding of a fundamental building block is studied in relation to iso-length spectrality of the corresponding domains. Although an explicit and exact trace formula such as…
We give a number of examples of isospectral pairs of plane domains, and a particularly simple method of proving isospectrality. One of our examples is a pair of domains that are not only isospectral but homophonic: Each domain has a…
Given a strictly convex domain $\Omega$ in $\R^2$, there is a natural way to define a billiard map in it: a rectilinear path hitting the boundary reflects so that the angle of reflection is equal to the angle of incidence. In this paper we…
The Laplace-Beltrami eigenfunctions on a compact Riemannian manifold $M$ whose geodesic billiard flow has mixed character have been conjectured by Percival to split into two complementary families, with all semiclassical mass supported in…
It is well known that certain pairs of planar domains have the same spectra of the Laplacian operator. We prove that these domains are still isospectral for a wider class of physical problems, including the cases of heterogeneous drums and…
This paper reports the results of extensive numerical studies related to spectral properties of the Laplacian and the scattering matrix for planar domains (called billiards). There is a close connection between eigenvalues of the billiard…
Properties of a quantum mushroom billiard in the form of a superconducting microwave resonator have been investigated. They reveal unexpected nonuniversal features such as, e.g., a supershell effect in the level density and a dip in the…
We prove that a a strongly convex planar domain (Birkhoff table) with dihedral symmetry, which is sufficiently close in a finitely smooth topology to an ellipse, is deformationally spectrally rigid within the class of domains preserving…
Concerning the Laplace operator with homogeneous Dirichlet boundary conditions, the classical notion of isospectrality assumes that two domains are related when they give rise to the same spectrum. In two dimensions, non isometric,…
Several examples of pairs of isospectral planar domains have been produced in the two-dimensional Euclidean space by various methods. We show that all these examples rely on the symmetry between points and blocks in finite projective…
In a celebrated paper '"Can one hear the shape of a drum?"' M. Kac [Amer. Math. Monthly 73, 1 (1966)] asked his famous question about the existence of nonisometric billiards having the same spectrum of the Laplacian. This question was…
A statistical analysis of the eigenfrequencies of two sets of superconducting microwave billiards, one with mushroom-like shape and the other from the familiy of the Limacon billiards, is presented. These billiards have mixed…
In this article we discuss pointwise spectral rigidity results for several billiard systems (e.g., Birkhoff billiards, symplectic billiards and $4$-th billiards), showing that a single value of Mather's $\beta$-function can determine…
Recently, the nodal domain counts of planar, integrable billiards with Dirichlet boundary conditions were shown to satisfy certain difference equations in [Ann. Phys. 351, 1-12 (2014)]. The exact solutions of these equations give the number…
We prove that equivalence classes of marked length isospectral Birkhoff billiard tables are compact in the $C^\infty$ topology, analogous to the Laplace spectral results of Melrose, Osgood, Phillips and Sarnak. To do so, we derive a…
We consider the class of dispersing billiard systems in the plane formed by removing three convex analytic scatterers satisfying the non-eclipse condition. The collision map in this system is conjugated to a subshift, providing a natural…
Symplectic billiards were introduced by Albers and Tabachnikov as billiards in strictly convex bounded domains of the plane with smooth boundary having a specific law of reflection. This paper proves a rigidity result for symplectic…
We develop a theory of planar, origin-symmetric, convex domains that are inextensible with respect to lattice covering, that is, domains such that augmenting them in any way allows fewer domains to cover the same area. We show that…
Eigenfunctions of integrable planar billiards are studied - in particular, the number of nodal domains, $\nu$, of the eigenfunctions are considered. The billiards for which the time-independent Schr\"odinger equation (Helmholtz equation) is…