Analytical and Computational Methods in Scattering and Applied Mathematics

Fadil Santosa, Ivar Stakgold

• Publisher: Chapman and Hall/CRC
• Year: 2019
• Print ISBN: 9781584881599
• eBook ISBN: 9780429525087
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Professor Ralph Kleinman was director of the Center for the Mathematics of Waves and held the UNIDEL Professorship of the University of Delaware. Before his death in 1998, he made major scientific contributions in the areas of electromagnetic scattering, wave propagation, and inverse problems. He was instrumental in bringing together the mathematic

Spectrum of Multidimensional Schrödinger Operators with Sparse Potentials

S. Molchanov^* B. Vainberg^†

Dedicated to the memory of Ralph Kleinman, a most remarkable person and dear friend.

Abstract

The paper contains analysis of the spectrum of multidimensional Schrödinger operators on the lattice ℤ^d and in ${\Re }^d$ with sparse potentials. The latter means that the potential is an infinite sum of bumps with distances between their supports going to infinity or with the density of the bumps decaying at infinity. Among other results, the coexistence of absolutely continuous and pure point spectrum is shown and an analog of Kato theorem is proven when potential decays at infinity only in average. It is also shown that in this case, there may exist eigenvalues embedded into continuous spectrum.

1 Introduction

This article is a continuation and a development of our recent papers [11], [13] on multiscattering by an infinite system of more and more distant (i.e. sparse) bumps. The central motivation of the work goes to the physics of disordered states. The fundamental (still unsolved) problem in this area is the existence of t...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• Optimization of Radiating Fields and the Example of Null-Placement
• Modified Gradient and Contrast Source Inversion
• Identification of Acoustically Soft Solids of Revolution in a Waveguide Using the ICBA Method
• Recollections of Ralph Kleinman
• An Asymptotic Wave Interpretation of Sonar Reflections from a Rough Surface
• Critical Values in Inverse Scattering Theory
• The Finite Element Approximation of Scattering in a Layered Medium
• The Far-Field Expansion Theorem in Thermoelastic Scattering
• Theoretical Notes on Scattering from One-Dimensional Penetrable Rough Surfaces
• Higher-Order Slender Body Theory
• Acoustic Scattering in Dispersions
• Applied Analysis in Ship Hydrodynamics
• Mathematical Foundations for the Boundary-Field Equation Methods in Acoustic and Electromagnetic Scattering
• Obstacle Scattering as Seen by R.K. and R.K.
• On Nonlinearized Wavefield Inversion Methods and the Identification of Buried Objects
• Scattering in a 2-D Optical Waveguide
• Scattering by Inclusions
• Pointwise Interpolation Inequalities for Riesz and Bessel Potentials
• Spectrum of Multidimensional Schrödinger Operators with Sparse Potentials
• On Inverse Problems for the Wave Equation with Time Dependent Potentials
• Photo Album