We prove a complex and a real interpolation theorems on Besov spaces and Triebel-Lizorkin spaces associated with a selfadjoint operator L, without assuming the gradient estimate for its spectral kernel. The result applies to the cases where L is a uniformly elliptic operator or a Schrödinger operator with electromagnetic potential.

The Heun equation can be rewritten as an eigenvalue equation for an ordinary differential operator of the form −d 2 /dx 2 + V (g; x), where the potential is an elliptic function depending on a coupling vector g ∈ R 4. Alternatively, this operator arises from the BC 1 specialization of the BC N elliptic nonrelativistic Calogero–Moser system (a.k.a. the Inozemtsev system). Under suitable restrict...

Abstract. In this article we introduce an asymptotic preserving scheme designed to compute the solution of a two dimensional elliptic equation presenting large anisotropies. We focus on an anisotropy aligned with one direction, the dominant part of the elliptic operator being supplemented with Neumann boundary conditions. A new scheme is introduced which allows an accurate resolution of this el...

A new adaptive Fundamental Solution Method (FSM) for the approximate solution of scalar elliptic boundary value problems is presented. The construction of the basis functions is based on the Adaptive Cross Approximation (ACA) of the fundamental solution of the corresponding elliptic operator. An algorithm for an immediate computer implementation of the method is formulated. A series of numerica...

We show how to parameterise a homogenised conductivity in R2 by a scalar function s(x), despite the fact that the conductivity parameter in the related up-scaled elliptic operator is typically tensor valued. Ellipticity of the operator is equivalent to strict convexity of s(x), and with consideration to mesh connectivity, this equivalence extends to discrete parameterisations over triangulated ...

We address a Cauchy problem for a nonlinear elliptic PDE arising in glaciology. After recasting the Cauchy problem as an ill-posed operator equation, we prove (for values of a certain parameter allowing Hilbert space techniques) differentiability properties of the associated operator. We also suggest iterative methods which can be applied to solve the operator problem.

In this paper we study the asymptotic behavior of the ground state energy E(R) of the Schrödinger operator PR = −∆ + V1(x) + V2(x−R), x, R ∈ IR, where the potentials Vi are small perturbations of the Laplacian in IR, n ≥ 3. The methods presented here apply also in the investigation of the ground state energy E(g) of the operator Pg = P + V1(x) + V2(gx), x ∈ X, g ∈ G, where Pg is an elliptic ope...

Abstract We demonstrate, through separation of variables and estimates from the semiclassical analysis of the Schrödinger operator, that the eigenvalues of an elliptic operator defined on a compact hypersurface in Rn can be found by solving an elliptic eigenvalue problem in a bounded domain Ω ⊂ Rn. The latter problem is solved using standard finite element methods on the Cartesian grid. We also...

A boundary value problem for a fractional power 0 < ε < 1 of the second-order elliptic operator is considered. The boundary value problem is singularly perturbed when ε→ 0. It is solved numerically using a time-dependent problem for a pseudo-parabolic equation. For the auxiliary Cauchy problem, the standard two-level schemes with weights are applied. The numerical results are presented for a mo...

Introduction. For a closed, densely defined linear operator T in a Hubert space H, we define the essential spectrum ess sp T as the complement in C of the set of X for which T X is a Fredholm operator (with possibly nonzero index). Recall (cf. Wolf [7] ) that X G ess sp T if and only if either T — X or 7* — X has a singular sequence, i.e. a sequence ukGH with \\uk\\ = 1 for all k, (T X)uk —> 0 ...