Semi‐Classical Asymptotic of Spectral Function for Some Schrödinger Operators

Abstract

In this paper one obtains a result concerning the asymptotic behaviour of the spectral function on the diagonal for SCHRODINOER operators Ah = --A + V as h -+ 0. This asymptotic change the form on the energy level V ( x ) = A. h2 2

A novel algorithm for image representation using discrete spectrum of the Schrödinger operator

“…We will obtain the proof by using a suitable extension of Karadzhov's theorem [13] on the spectral function (Theorem Appendix A.1) and some Riesz means connected to a Lieb-Thirring's conjecture proposed by Helffer and Robert [11] (Theorem Appendix A.2). …”

“…The proof of this result is obtained using the generalization of Theorem 4.1, proposed by Helffer and Laleg in [10], to 2D case, which uses an extension of Karadzhov's theorem on the spectral function [13], together with the connection of the Riesz means with Lieb-Thirring conjecture proposed by Helffer and Robert in [11]. Details of the proof are provided in Appendix A.…”

“…Then, inspired form the semi-classical properties of the Schrö-dinger operator [11,13], the extension of the SCSA formula in 2D case is given by the following theorem. Theorem 3.1.…”

“…The next theorem is a generalization of 1D to 2D, of Theorem 4.1 proposed by Helffer and Laleg in [10], which is a suitable extension of Karadzhov's Theorem on the spectral function [13]. …”

“…The reconstruction formula is first derived for the 2D case thanks to some concepts from the semi-classical analysis of the Schrödinger operator [11,13]. Then an efficient algorithm is proposed where the two dimensional (2D) semi-classical Schrödinger operator is splited into two one dimensional (1D) operators and then the squared L 2 -normalized eigenfunctions of these 1D operators are combined using a tensor product approach [8,12].…”