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Advanced Textbooks in Mathematics · / _The /1uner / storm \ \ 1 Wiiher Transform Advanced Textbooks in Mathematics ISSN: 2059-769X Published The Wigner Transform by Maurice de Gosson Forthcoming Periods and Special Functions in Transcendence by Paula B Tretkoff Conformal Maps and Geometry by Dmitry Belyaev Advanced Textbooks in Mathematics _The IUDer Transform Maurice de Gosson University of Vienna, Austria r t t. --l ·-f· < -~· ·--· -·~· ·- ·:f--'·:~~~-.-_.._ }?J\ / \ \ 1 : ~ .1 ·; i ,,':·.,.-_·_- _-_-_-_- _ ;· \. '~ World Scientific EW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO Published by World Scientific Publishing Europe Ltd. 57 Shelton Street, Covent Garden, London WC2H 9HE Head office: 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 Library of Congress Cataloging-in-Publication Data Names: Gosson, Maurice de. Title: The Wigner transform I by Maurice de Gosson (University of Vienna, Austria). Description: New Jersey : World Scientific, 2017. I Series: Advanced textbooks in mathematics Identifiers: LCCN 20160566801 ISBN 9781786343086 (he : alk. paper) I ISBN 9781786343093 (pbk : alk. paper) Subjects: LCSH: Phase space (Statistical physics) I Statistical mechanics. Classification: LCC QC174.85.P48 G67 20171 DDC 530.15/95--dc23 LC record available at https://lccn.loc.gov/2016056680 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2017 by World Scientific Publishing Europe Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission ji'Om the Publisha For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Desk Editors: V. Vishnu Mohan/Mary Simpson Typeset by Stallion Press Email: [email protected] Printed in Singapore To My Charlyne, With All My Love Preface The study of the Wigner transform and of its twin brother, the ambiguity function, has an already long and illustrious history, with strong links to both pure and applied mathematics, and to physics and engineering. In this book, we aim at giving a comprehensive and mathematically rigorous treatment of these subjects at a level which should be accessible to undergraduate upper~level students in mathematics and mathematical physics (we have put a special emphasize on the applications to quantum mechanics, some of them of a rather advanced level). The material should also be of interest to mathematicians and engineers working in signal analysis and time-frequency analysis; the confirmed researcher can use this book as a reference work, since we have tried to unify different topics which are more or less spread out in the literature. Needless to saY,' this book is not the first approaching the Wigner transform using methods from harmonic analysis. Well-cited predecessors are the comprehensive books by Folland Harmonic Analysis in Phase Space and by Grochenig Time-Frequency Analysis. The number of contributions in the form of research or expository papers defies the imagination: we have not been able to take into account all the recent advances, but rather focused on the potentially most important; of course these choices do reflect the author's own tastes. The book is structured as a series of Lecture Notes; the order in which . the topics are introduced is therefore linear. I have tried to present the material using the principle of parsimony, and the mathematics is presented vii viii Preface and structured following the old-fashioned but efficient way Definition - Lemma - Proposition - Corollary (including, of course, Examples and Remarks). The chapters are short, and designed for a 60 minutes lecture (this may be wishful thinking - it depends of course on the initial level of the audience; I have here in mind upper-level European or American undergraduates or graduate students). This work has been financed by the grant P27773-N23 of FWF Der Wissenschaftsfonds (the Austrian Research Agency) Maurice A. de Gosson Vienna, Summer 2016