A Dressing Method in Mathematical Physics
The monograph is devoted to the systematic presentation of the so called "dressing method" for solving differential equations (both linear and nonlinear) of mathematical physics. The essence of the dressing method consists in a generation of new non-trivi
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MATHEMATICAL PHYSICS STUDIES Editorial Board:
Maxim Kontsevich, IHES, Bures-sur-Yvette, France Massimo Porrati, New York, University, New York, U.S.A. Vladimir Matveev, Universit´e Bourgogne, Dijon, France Daniel Sternheimer, Universit´e Bourgogne, Dijon, France
VOLUME 28
A Dressing Method in Mathematical Physics by
Evgeny V. Doktorov Institute of Physics, Minsk, Belarus
and
Sergey B. Leble University of Technology, Gdansk, Poland
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4020-6138-7 (HB) ISBN 978-1-4020-6140-0 (e-book) Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com
Printed on acid-free paper
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55 udir convienmi ancor come l’essemplo 56 e l’essemplare non vanno d’un modo, 57 ch´e io per me indarno a ci` o contemplo. Dante Alighieri, Divina Commedia Paradiso, Canto XXVIII 55 then I still have to hear just how the model 56 and copy do not share in one same plan 57 for by myself I think on this in vain. Translated by A. Mandelbaum
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv 1
Mathematical preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Intertwining relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Ladder operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Definitions and Lie algebra interpretation . . . . . . . . . . . . 1.2.2 Hermitian ladder operators . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Jaynes–Cummings model . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Results for differential operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Commuting ordinary differential operators . . . . . . . . . . . . 1.3.2 Direct consequences of intertwining relations in the matrix case and multidimensions . . . . . . . . . . . . . . 1.4 Hyperspherical coordinate systems and ladder operators . . . . . . 1.5 Laplace transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Matrix factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 QR algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.3 Factorization of the λ matrix . . . . . . . . . . . . . . . . . . . . . . . 1.7 Elementary factorization of matrix . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Ma
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