Bound states of the magnetic Schrödinger operator
By: Nicolas Raymond
Series: EMS tracts in mathematics -- vol. 27; EMS tracts in mathematics -- 27.
Publisher: European Mathematical Society
Published: 2017
Place of publication: Zürich, Switzerland
Language: eng
Number of pages: 380
ISBN-13: 9783037191699
Subjects: Schrödinger operator; Spectral theory (Mathematics); Magnetism; SCIENCE / Energy; SCIENCE / Mechanics / General; SCIENCE / Physics / General
Description: This book is a synthesis of recent advances in the spectral theory of the magnetic Schrödinger operator. It can be considered a catalog of concrete examples of magnetic spectral asymptotics. Since the presentation involves many notions of spectral theory and semiclassical analysis, it begins with a concise account of concepts and methods used in the book and is illustrated by many elementary examples. Assuming various points of view (power series expansions, Feshbach-Grushin reductions, WKB constructions, coherent states decompositions, normal forms) a theory of Magnetic Harmonic Approximation is then established which allows, in particular, accurate descriptions of the magnetic eigenvalues and eigenfunctions. Some parts of this theory, such as those related to spectral reductions or waveguides, are still accessible to advanced students while others (e.g., the discussion of the Birkhoff normal form and its spectral consequences, or the results related to boundary magnetic wells in dimension three) are intended for seasoned researchers.
Notes: Includes bibliographical references and index.
Table of contents: title: Part 1 Methods and examples - Elements of Spectral Theory - Examples - First semiclassical examples - From local models to global estimates - Birkhoff normal form in dimension one - Part 2 Main theorems - Spectral reductions - Mangnetic wells in dimension two - Boundary magnetic wells in dimension three - Waveguides - On some connected non-linear problems - Part 3 Spectral reductions - Electric Born-Oppenheimer approximation - Magnetic Born-Oppenheimer approximation - Examples of magnetic WKB constructions - Part 4 Magnetic wells in dimension two - Vanishing magnetic fields in dimension two - Non-vanishing magnetic fields - Semiclassical non-linear magnetic eigenvalues - Part 5 Boundary magnetic wells in dimension three - Magnetic half-space - Magnetic wedge - Magnetic cone - Part 6 Waveguides - Magnetic effects in curved waveguides - Spectrum of thin triangles and broken waveguides - Non-linear dynamics in bidimensional waveguides.