We prove a 2-terms Weyl formula for the counting function N(mu) of the spectrum of the Laplace operator in the Euclidean disk with a sharp remainder estimate O(mu^2/3).
For the unit sphere S^d in Euclidean space R^(d+1), we show that for d-1<s<d and any N>1, discrete N-point minimal Riesz s-energy configurations are well separated in the sense that the minimal distance between any pair of distinct points in such a configuration is bounded below by C/N^(1/d), where C is a positive constant depending on s and d.
We explore an algorithm which systematically finds all discrete eigenvalues of an analytic eigenvalue problem. The algorithm is more simple and elementary as could be expected before. It consists of Hejhals identity, linearisation, and Turing bounds. Using the algorithm, we compute more than one hundredsixty thousand consecutive eigenvalues of the Laplacian on the modular surface, and investigate the asymptotic and statistic properties of the fluctuations in the Weyl remainder. We summarize the findings in two conjectures. One is on the maximum size of the Weyl remainder, and the other is on the distribution of a suitably scaled version of the Weyl remainder.
A version of Connes trace formula allows to associate a measure on the essential spectrum of a Schrodinger operator with bounded potential. In solid state physics there is another celebrated measure associated with such operators --- the density of states. In this paper we demonstrate that these two measures coincide. We show how this equality can be used to give explicit formulae for the density of states in some circumstances.
In a previous paper we introduced and developed a recursive construction of joint eigenfunctions $J_N(a_+,a_-,b;x,y)$ for the Hamiltonians of the hyperbolic relativistic Calogero-Moser system with arbitrary particle number $N$. In this paper we focus on the cases $N=2$ and $N=3$, and establish a number of conjectured features of the corresponding joint eigenfunctions. More specifically, choosing $a_+,a_-$ positive, we prove that $J_2(b;x,y)$ and $J_3(b;x,y)$ extend to globally meromorphic functions that satisfy various invariance properties as well as a duality relation. We also obtain detailed information on the asymptotic behavior of similarity transformed functions E$_2(b;x,y)$ and E$_3(b;x,y)$. In particular, we determine the dominant asymptotics for $y_1-y_2toinfty$ and $y_1-y_2,y_2-y_3toinfty$, resp., from which the conjectured factorized scattering can be read off.