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Konrad Schöbel

An Algebraic Geometric Approach to Separation of Variables

Konrad Schöbel Friedrich-Schiller-Universität Jena Germany Habilitationsschrift Friedrich-Schiller-Universität Jena, 2014

ISBN 978-3-658-11407-7 ISBN 978-3-658-11408-4 (eBook) DOI 10.1007/978-3-658-11408-4 Library of Congress Control Number: 2015949632 Springer Spektrum © Springer Fachmedien Wiesbaden 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speci¿cally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on micro¿lms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speci¿c statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer Spektrum is a brand of Springer Fachmedien Wiesbaden Springer Fachmedien Wiesbaden is part of Springer Science+Business Media (www.springer.com)

An Algebraic Geometric Approach to Separation of Variables

∞ 1

0 M¯0,4(R) ∼ = RP1

0

1 ∞

To Alba & Aitana

Preface Separation of variables is one of the oldest and most powerful methods to construct exact solutions of fundamental partial differential equations in classical and quantum physics, like the Hamilton-Jacobi equation in Newtonian mechanics, the Schr¨odinger equation in quantum mechanics or the wave equation. A separation of the Schr¨odinger equation for the hydrogen atom in spherical coordinates, for example, yields functions describing the orbital structure of the electrons, i.e. the basis for the periodic table of the elements, and is thus at the root of chemistry. Likewise, many other well known special functions which are used all over in science and technology stem from a separation of variables. The problem to classify all coordinate systems in which this method is applicable, the so-called separation coordinates, was solved exhaustively by Ernest G. Kalnins & Willard Miller Jr. over 30 years ago. For this reason many experts would consider the theory of separation of variables as settled or even old-fashioned. However, in this book we argue that the above classification problem is essentially an algebraic geometric and not a differential geometric problem, i.e. governed by algebraic instead of partial differential equations. This means that Kalnins & Mi