%0 Journal Article
%T A Survey of Some Topics Related to Differential Operators
%A Denise Huet
%J Journal of Operators
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/562963
%X This paper is the result of investigations suggested by recent publications and completes the work of Huet, 2010. The topics, which are dealt with, concern some spaces of functions and properties of solutions of linear and nonlinear, stationary and evolution differential equations, namely, existence, spectral properties, resonances, singular perturbations, boundary layers, and inertial manifolds. They are presented in the alphabetical order. The aim of this document and of Huet, 2010, is to be a useful reference for (young) researchers in mathematics and applied sciences. 1. Introduction The article is divided into several sections entitled: Birman-Schwinger operators; BMO spaces; Bounded variation (functions of); Discrete energy; Dissipative operators; Dynamical systems; Equal- area condition; Inertial manifolds; Mathieu-Hill type equations; Memory (equations with); Nodes, Nodal; Resonances. The development of each entry includes indications on history, definitions, an overview of main results, examples, and applications but is, of course, nonexhaustive. Complements will be found in the references. A prepublication of some entries is presented in Huet [1]. 2. Birman-Schwinger Operator Definition 1. Consider the Schr£¿dinger operator acting on , , where is a real-valued continuous function defined on which is nonnegative and tends to zero, sufficiently fast, as £¿£¿and£¿£¿ is a small negative coupling constant. The operator is self-adjoint and its spectrum is . The Birman-Schwinger operator associated with (1) is the operator where is the resolvent of in . For each ,£¿£¿ is self-adjoint and compact (cf. Arazy and Zelenko [2]). Application. In [2], the authors consider the decomposition , where is a finite rank operator and an Hilbert-Schmidt operator whose norm is uniformly bounded with respect to for some . An asymptotic expansion of the bottom virtual eigenvalue of , as tends to zero, is deduced from this decomposition: if is odd, it is of power type, while, when is even, it involves the log function. Asymptotic estimates are obtained, as , for the nonbottom virtual eigenvalues of , , where if is odd and if is even. If is odd, is a meromorphic operator function, and the leading terms of the asymptotic estimates of are of power type. An algorithm, based on the Puiseux-Newton diagram (cf. Baumg£¿rtel [3]), is proposed for an evaluation of the leading coefficients of these estimates. If d is even, two-sided estimates are obtained for eigenvalues with an exponential rate of decay; the rest of the eigenvalues have a power rate of decay. Estimates of Lieb-Thirring
%U http://www.hindawi.com/journals/joper/2013/562963/