Differences Between Robin and Neumann Eigenvalues.

Let Ω âŠ‚ R 2 be a bounded planar domain, with piecewise smooth boundary ∂ Ω . For σ > 0 , we consider the Robin boundary value problem - Δ f = λ f , ∂ f ∂ n + σ f = 0 on ∂ Ω where ∂ f ∂ n is the derivative in the direction of the outward pointing normal to ∂ Ω . Let 0 < λ 0 σ ≤ λ 1 σ ≤ … be the corresponding eigenvalues. The purpose of this paper is to study the Robin-Neumann gaps d n ( σ ) : = λ n σ - λ n 0 . For a wide class of planar domains we show that there is a limiting mean value, equal to 2 length ( ∂ Ω ) / area ( Ω ) · σ and in the smooth case, give an upper bound of d n ( σ ) ≤ C ( Ω ) n 1 / 3 σ and a uniform lower bound. For ergodic billiards we show that along a density-one subsequence, the gaps converge to the mean value. We obtain further properties for rectangles, where we have a uniform upper bound, and for disks, where we improve the general upper bound.

Volume distribution of nodal domains of random band-limited functions.

We study the volume distribution of nodal domains of families of naturally arising Gaussian random fields on generic manifolds, namely random band-limited functions. It is found that in the high energy limit a typical instance obeys a deterministic universal law, independent of the manifold. Some of the basic qualitative properties of this law, such as its support, monotonicity and continuity of the cumulative probability function, are established.

On probability measures arising from lattice points on circles.

A circle, centered at the origin and with radius chosen so that it has non-empty intersection with the integer lattice Z 2 , gives rise to a probability measure on the unit circle in a natural way. Such measures, and their weak limits, are said to be attainable from lattice points on circles. We investigate the set of attainable measures and show that it contains all extreme points, in the sense of convex geometry, of the set of all probability measures that are invariant under some natural symmetries. Further, the set of attainable measures is closed under convolution, yet there exist symmetric probability measures that are not attainable. To show this, we study the geometry of projections onto a finite number of Fourier coefficients and find that the set of attainable measures has many singularities with a "fractal" structure. This complicated structure in some sense arises from prime powers-singularities do not occur for circles of radius n if n is square free.