Exponentially Small Asymptotic Formulas for the Length Spectrum in Some Billiard Tables

Some Asymptotic Formulas for Multiplicative Functions

On Marked Length Spectrums of Generic Strictly Convex Billiard Tables

In this paper we show that for a generic strictly convex domain, one can recover the eigendata corresponding to Aubry-Mather periodic orbits of the induced billiard map, from the (maximal) marked length spectrums of the domain.

Some Asymptotic Formulas for Wiener Integralso

Small Gaps in the Spectrum of the Rectangular Billiard

We study the size of the minimal gap between the first N eigenvalues of the Laplacian on a rectangular billiard having irrational squared aspect ratio α, in comparison to the corresponding quantity for a Poissonian sequence. If α is a quadratic irrationality of certain type, such as the square root of a rational number, we show that the minimal gap is roughly of size 1/N , which is essentially ...

Search Light in Billiard Tables

We investigate whether a search light, S, illuminating a tiny angle (“cone”) with vertex A inside a bounded region Q ∈ IR with the mirror boundary ∂Q, will eventually illuminate the entire region Q. It is assumed that light rays hitting the corners of Q terminate. We prove that: (I) if Q = a circle or an ellipse, then either the entire Q or an annulus between two concentric circles/confocal ell...