相关论文: Patterson-Sullivan distributions of finite regular…

We establish conditions for which graph Laplacians $\Delta_{\lambda,\epsilon}$ on compact, boundaryless, smooth submanifolds $\mathcal{M}$ of Euclidean space are semiclassical pseudodifferential operators ($\Psi$DOs): essentially, that the…

For a finite graph, we establish natural isomorphisms between eigenspaces of a Laplace operator acting on functions on the edges and eigenspaces of a transfer operator acting on functions on one-sided infinite non-backtracking paths.…

We analyze quasi probability distributions in discrete phase space related to the discrete Heisenberg-Weyl group. In particular, we discuss the relation between the Discrete Wigner and Q- functions.

In graph signal processing, the graph adjacency matrix or the graph Laplacian commonly define the shift operator. The spectral decomposition of the shift operator plays an important role in that the eigenvalues represent frequencies and the…

In this paper we study Lipschitz regularity of elliptic PDEs on geometric graphs, constructed from random data points. The data points are sampled from a distribution supported on a smooth manifold. The family of equations that we study…

We introduce the discrete poly-Laplace operator on a subgraph with Dirichlet boundary condition. We obtain upper and lower bounds for the sum of the first $k$ Dirichlet eigenvalues of the poly-Laplace operators on a finite subgraph of…

It has been suggested that the distribution of the suitably normalized number of zeros of Laplacian eigenfunctions contains information about the geometry of the underlying domain. We study this distribution (more precisely, the…

In this paper we give an overview over some aspects of the modern mathematical theory of Ruelle resonances for chaotic, i.e. uniformly hyperbolic, dynamical systems and their implications in physics. First we recall recent developments in…

We establish several new relations between the discrete transition operator, the continuous Laplacian and the averaging operator associated with combinatorial and metric graphs. It is shown that these operators can be expressed through each…

We develop a Logvinenko--Sereda theory for one-dimensional vector-valued self-adjoint operators. We thus deliver upper bounds on $L^2$-norms of eigenfunctions -- and linear combinations thereof -- in terms of their $L^2$- and…

In this paper we study the common distance between points and the behavior of a constant length step discrete random walk on finite area hyperbolic surfaces. We show that if the second smallest eigenvalue of the Laplacian is at least 1/4,…

We present some new results on the joint distribution of an arbitrary subset of the ordered eigenvalues of complex Wishart, double Wishart, and Gaussian hermitian random matrices of finite dimensions, using a tensor pseudo-determinant…

We show that the Riemannian Gaussian distributions on symmetric spaces, introduced in recent years, are of standard random matrix type. We exploit this to compute analytically marginals of the probability density functions. This can be done…

We construct quantum models of two particles on a compact metric graph with singular two-particle interactions. The Hamiltonians are self-adjoint realisations of Laplacians acting on functions defined on pairs of edges in such a way that…

The purpose of this paper is to study transport equations on the unit tangent bundle of closed oriented Riemannian surfaces and to connect these to the transport twistor space of the surface (a complex surface naturally tailored to the…

We investigate the level spacing distribution for the quantum spectrum of the square billiard. Extending work of Connors--Keating, and Smilansky, we formulate an analog of the Hardy--Littlewood prime $k$-tuple conjecture for sums of two…

In this note, we prove a theorem \`a la Fatou for the square root of Poisson Kernel in the context of quasi-convex cocompact discrete groups of isometries of $\delta$-hyperbolic spaces. As a corollary we show that some matrix coefficients…

We obtain the limiting distribution of the nodal area of random Gaussian Laplace eigenfunctions on $\mathbb{T}^3= \mathbb{R}^3/ \mathbb{Z}^3$ ($3$-dimensional 'arithmetic random waves'). We prove that, as the multiplicity of the eigenspace…

We study the asymptotic behaviour of eigenvalues and eigenmodes of the Witten Laplacian on a smooth compact Riemannian manifold without boundary. We show that they converge to the Pollicott-Ruelle spectrum of the corresponding gradient flow…

We explore the limiting empirical eigenvalue distributions arising from matrices of the form \[A_{n+1} = \begin{bmatrix} A_n & I\\ I & A_n \end{bmatrix} , \]where $A_0$ is the adjacency matrix of a $k$-regular graph. We find that for…