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. ...

Abstract. In the objective of studying concentration and oscillation properties of eigenfunctions of the discrete Laplacian on regular graphs, we construct a pseudo-differential calculus on homogeneous trees, their universal covers. We define symbol classes and associated operators. We prove that these operators are bounded on L and give adjoint and product formulas. Finally we compute the symb...

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. ...

In this article, we focus on the analysis of discrete versions of the Calderón problem in dimension d ≥ 3. In particular, our goal is to obtain stability estimates for the discrete Calderón problems that hold uniformly with respect to the discretization parameter. Our approach mimics the one in the continuous setting. Namely, we shall prove discrete Carleman estimates for the discrete Laplace o...

In the first part of the article we introduce C-algebras associated to self-similar groups and study their properties and relations to known algebras. The algebras are constructed as sub-algebras of the Cuntz-Pimsner algebra (and its homomorphic images) associated with the self-similarity of the group. We study such properties as nuclearity, simplicity and Morita equivalence with algebras relat...

An electrical resistance network (ERN) is a connected graph (G, c). The conductance function cxy weights the edges, which are then interpreted as resistors of possibly varying strengths. The relationship between the natural Dirichlet form E and the discrete Laplace operator ∆ on a finite network is given by E(u, v) = 〈u,∆v〉2 , where the latter is the usual l2 inner product. We extend this formu...

Let Ω be a convex domain with smooth boundary in Rd. It has been shown recently that the semigroup generated by the discrete Laplacian for quasi-uniform families of piecewise linear finite element spaces on Ω is analytic with respect to the maximum-norm, uniformly in the mesh-width. This implies a resolvent estimate of standard form in the maximum-norm outside some sector in the right halfplane...

We continue our study of intermittency for the parabolic Anderson equation ∂u/∂t = κΔu + ξu, where u : Z × [0,∞) → R, κ is the diffusion constant, Δ is the discrete Laplacian, and ξ : Z × [0,∞) → R is a space-time random medium. The solution of the equation describes the evolution of a “reactant” u under the influence of a “catalyst” ξ. In this paper we focus on the case where ξ is exclusion wi...

We compute the asymptotic determinant of the discrete Laplacian on a simply-connected rectilinear region in R2. Specifically, for each > 0 let H be the subgraph of Z2 whose vertices lie in a fixed rectilinear polygon U . Let N (H ) denote the number of vertices of H and B(H ) the number of vertices on the boundary (the outer face). Then the log of the determinant of the Laplacian on H has the f...

Using a general approximation setting having the generic properties of finite-elements, we prove uniform boundedness and stability estimates on the discrete Stokes operator in Sobolev spaces with fractional exponents. As an application, we construct approximations for the timedependent Stokes equations with a source term in L(0, T ;L(Ω)) and prove uniform estimates on the time derivative and di...