Consider the centered Gaussian field on the lattice Z, d large enough, with covariances given by the inverse of ∑K j=k qj(−∆) , where ∆ is the discrete Laplacian and qj ∈ R, k ≤ j ≤ K, the qj satisfying certain additional conditions. We extend a previously known result to show that the probability that all spins are nonnegative on a box of side-length N has an exponential decay at rate of order...

We prove an analogue of a classical asymptotic stability result of standing waves of the Schrödinger equation originating in work by Soffer and Weinstein. Specifically, our result is a transposition on the lattice Z of a result by Mizumachi [M1] and it involves a discrete Schrödinger operator H = −∆+q. The decay rates on the potential are less stringent than in [M1], since we require q ∈ l1,1. ...

We prove Lp-bounds on the Fourier transform of measures μ supported on two dimensional surfaces. Our method allows to consider surfaces whose Gauss curvature vanishes on a one-dimensional submanifold. Under a certain non-degeneracy condition, we prove that μ̂ ∈ L4+β , β > 0, and we give a logarithmically divergent bound on the L4-norm. We use this latter bound to estimate almost singular integra...

In this paper we design a smoothing filter for texture+depth images based on anisotropic diffusion. Our proposed filter is linear and allows to generate a scale space of the texture image which preserves internal image structure (i.e. it does not smooth through object boundaries). Moreover, we show experimentally that our filter demonstrates better stability under camera position changes in suc...

We prove an analogue of a classical asymptotic stability result of standing waves of the Schrödinger equation originating in work by Soffer and Weinstein. Specifically, our result is a transposition on the lattice Z of a result by Mizumachi [M1] and it involves a discrete Schrödinger operator H = −∆+q. The decay rates on the potential are less stringent than in [M1], since we require q ∈ l1,1. ...

The present note reviews recent results on resonances for one-dimensional quantum ergodic systems constrained to a large box. We restrict ourselves to one dimensional models in the discrete case. We consider two type of ergodic potentials on the half-axis, periodic potentials and random potentials. For both models, we describe the behavior of the resonances near the real axis for a large typica...

The discrete spectrum of complex Jacobi matrices that are compact perturbations of the discrete laplacian is under consideration. The rate of stabilization for the matrix entries sharp in the sense of order which provides finiteness of the discrete spectrum is found. The Jacobi matrix with the discrete spectrum having the only limit point is constructed. The results can be viewed as the discret...

Discrete Green’s functions are the inverses or pseudo-inverses of combinatorial Laplacians. We present compact formulas for discrete Green’s functions, in terms of the eigensystems of corresponding Laplacians, for products of regular graphs with or without boundary. Explicit formulas are derived for the cycle, torus, and 3-dimensional torus, as is an inductive formula for the t-dimensional toru...

Let Z + = Z d × Z+, let H0 be the discrete Laplacian on the Hilbert space l2(Z + ) with a Dirichlet boundary condition, and let V be a potential supported on the boundary ∂Z + . We introduce the notions of surface states and surface spectrum of the operator H = H0 + V and explore their properties. Our main result is that if the potential V is random and if the disorder is either large or small ...

There has been a long-standing question of whether certain mesh restrictions are required for a maximum condition to hold for the discrete equations arising from a finite element approximation of an elliptic problem. This is related to knowing whether the discrete Green’s function is positive for triangular meshes allowing sufficiently good approximation of H1 functions. We study this question ...