Result: Semiclassical Low Energy Scattering for One-Dimensional Schrodinger Operators with Exponentially Decaying Potentials

Title:
Semiclassical Low Energy Scattering for One-Dimensional Schrodinger Operators with Exponentially Decaying Potentials
Authors:
COSTIN, Ovidiu, DONNINGER, Roland, SCHLAG, Wilhelm, TANVEER, Saleh
Source:
Annales Henri Poincaré. 13(6):1371-1426
Publisher Information:
Heidelberg: Springer, 2012.
Publication Year:
2012
Physical Description:
print, 22 ref
Original Material:
INIST-CNRS
Subject Terms:
Mathematics, Mathématiques, Physics, Physique, Theoretical physics, Physique théorique, Sciences exactes et technologie, Exact sciences and technology, Sciences et techniques communes, Sciences and techniques of general use, Mathematiques, Mathematics, Analyse mathématique, Mathematical analysis, Equations aux dérivées partielles, Partial differential equations, Physique, Physics, Generalites, General, Méthodes mathématiques en physique, Mathematical methods in physics, Divers, Other topics in mathematical methods in physics, Calcul 1 dimension, One-dimensional calculations, En ligne, On line, En línea, Energie, Energy, Equation onde, Wave equations, Estimation erreur, Error estimation, Estimación error, Flot, Flow, Oleada, Matrice H, H matrix, Matriz H, Matrice diffusion, Scattering matrix, Matriz difusión, Opérateur Schrödinger, Schrödinger operator, Operador Schrodinger, Opérateur diffusion, Scattering operator, Operador difusión, Physique mathématique, Mathematical physics, Potentiel, Potentials, Représentation, Representation, Representación, Solution globale, Global solution, Solución global, Symétrie, Symmetry, 35J05, 35J10, H-matrice, Mesure spectrale, Opérateur potentiel
Document Type:
Academic journal Article
File Description:
text
Language:
English
Author Affiliations:
Department of Mathematics The Ohio State University 100 Math Tower 231 West 18th Avenue, Columbus, OH 43210-1174, United States
Ecole Polytechnique Fédérale de Lausanne MA B1 487, Station 8, 1015 Lausanne, Switzerland
Department of Mathematics University of Chicago 5734 South University Avenue, Chicago, IL 60637, United States
ISSN:
1424-0637
Access URL:
http://pascal-francis.inist.fr/vibad/index.php?action=search&terms=26256332
Rights:
Copyright 2015 INIST-CNRS
CC BY 4.0
Sauf mention contraire ci-dessus, le contenu de cette notice bibliographique peut être utilisé dans le cadre d’une licence CC BY 4.0 Inist-CNRS / Unless otherwise stated above, the content of this bibliographic record may be used under a CC BY 4.0 licence by Inist-CNRS / A menos que se haya señalado antes, el contenido de este registro bibliográfico puede ser utilizado al amparo de una licencia CC BY 4.0 Inist-CNRS
Notes:
Mathematics

Theoretical physics
Accession Number:
edscal.26256332
Database:
PASCAL Archive

Further Information

We consider semiclassical Schrödinger operators on the real line of the form H(ħ) = ―ħ2 / d2 dx2+ V(·; ħ) with ħ > 0 small. The potential V is assumed to be smooth, positive and exponentially decaying towards infinity. We establish semiclassical global representations of Jost solutions f± (·, E; ħ) with error terms that are uniformly controlled for small E and ħ, and construct the scattering matrix as well as the semiclassical spectral measure associated with H(ħ). This is crucial in order to obtain decay bounds for the corresponding wave and Schrödinger flows. As an application we consider the wave equation on a Schwarzschild background for large angular momenta ℓ were the role of the small parameter ħ is played by ℓ―1. It follows from the results in this paper and Donninger et al. (Commun Math Phys 2009, ar-Xiv:0911.3179), that the decay bounds obtained in Donninger et al. (Adv Math 226(1):484―540, 2011) and Donninger and Wilhelm (Int Math Res Not IMRN 22:4276―4300, 2010) for individual angular momenta ℓ can be summed to yield the sharp t―3 decay for data without symmetry assumptions.