Kuru, S.

Negro, J.

Nieto Calzada, Luis Miguel

English

Negro Vadillo, Francisco Javier

Repositorio Documental de la Universidad de Valladolid

CRM Series in Mathematical PhysicsŞengül KuruJavier NegroLuis M. Nieto    Editors Integrability, Supersymmetry and Coherent StatesA Volume in Honour of Professor Véronique HussinUNCORRECTEDPROOFCRM Series in Mathematical Physics 12Editorial Board: 3Joel S. Feldman 4Department of Mathematics 5University of British Columbia 6Vancouver, British Columbia V6T 1Z2 7Canada 8feldman@math.ubc.ca 910Duong H. Phong 11Department of Mathematics 12Columbia University 13New York, NY 10027–0029 14USA 15phong@math.columbia.edu 1617Yvan Saint-Aubin 18Département de Mathématiques 19et de Statistique 20Université de Montréal 21C.P. 6128, Succursale Centre-ville 22Montréal, Québec H3C 3J7 23Canada 24saint@dms.umontreal.ca 2526Luc Vinet 27Département de Mathématiques 28et de Statistique 29CRM, Université de Montréal 30C.P. 6128, Succursale Centre-ville 31Montréal, Québec H3C 3J7 32Canada 33vinet@crm.umontreal.ca 34UNCORRECTEDPROOFMore information about this series at http://www.springer.com/series/3872 35UNCORRECTEDPROOFS¸engu¨l Kuru • Javier Negro • Luis M. Nieto 36Editors 37Integrability, Supersymmetryand Coherent States3839A Volume in Honour of Professor VéroniqueHussin404112342UNCORRECTEDPROOFEditorsS¸engu¨l KuruDepartment of PhysicsAnkara UniversityAnkara, TurkeyJavier NegroDepartment of Theoretical, Atomicand Optical PhysicsUniversity of ValladolidValladolid, SpainLuis M. NietoDepartment of Theoretical, Atomicand Optical PhysicsUniversity of ValladolidValladolid, Spain43CRM Series in Mathematical PhysicsAQ1ISBN 978-3-030-20086-2 ISBN 978-3-030-20087-9 (eBook)https://doi.org/10.1007/978-3-030-20087-944© Springer Nature Switzerland AG 2019 45This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting, reproduction on microfilms or in any other physical way, and transmission or informationstorage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodologynow known or hereafter developed.4647484950The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.515253The publisher, the authors, and the editors are safe to assume that the advice and information in this bookare believed to be true and accurate at the date of publication. Neither the publisher nor the authors orthe editors give a warranty, express or implied, with respect to the material contained herein or for anyerrors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaims in published maps and institutional affiliations.5455565758This Springer imprint is published by the registered company Springer Nature Switzerland AG. 59The registered company address is: Gewerbestrasse 11, 6330 Cham, SwitzerlandUNCORRECTEDPROOFForeword 1The 6th International Workshop on New Challenges in Quantum Mechanics: 2Integrability and Supersymmetry was organized in Valladolid (Spain) on June 27– 330, 2017, in honor of Prof. Véronique Hussin. The conference was very successful: 4it attracted more than 60 researchers from different countries, and about 45 talks 5were delivered.AQ2 6Prof. Hussin is an internationally recognized expert in many branches of 7mathematical physics. Her career started in her native city of Liège (Belgium); 8afterward she moved to Montréal, where she has stayed ever since at Université de 9Montréal and at CRM (Centre de Recherches Mathématiques). However, Véronique 10Hussin has had numerous stays in different research centers collaborating with 11many colleagues and friends. In particular, we want to mention her regular visits 12to Durham University (England), CINVESTAV (Mexico), and the University of 13Valladolid (Spain) where, besides her scientific activity, due to her open character, 14she made many friends. She has made remarkable contributions to several branches 15of mathematical physics, like coherent and squeezed states, supersymmetry, sigma 16models, nonlinear equations, and other topics that show her wide range of interests. 17The workshop program payed special attention to the topics in which Prof. 18Hussin has been working. Based on this conference, but open to scientists in these 19areas, we considered a good opportunity to edit a contributed book published by 20Springer Nature, under the title “Integrability, Supersymmetry and Coherent States”. 21The book consists in a part devoted to review papers on the three main topics, whose 22authors are well-known experts in their fields and where the objective is to take a 23personal approach on some attractive aspects of these subjects. A second part is 24Special thanks are due to the University of Valladolid, where the meeting in honor of Prof.Véronique Hussin 6th International Workshop on New Challenges in Quantum Mechanics:Integrability and Supersymmetry was held.Partial financial support from Junta de Castilla y León and FEDER (Projects VA057U16,VA137G18, and BU229P18) and Ministerio de Economía y Competitividad of Spain (ProjectMTM2014-57129-C2-1-P) is acknowledged.vUNCORRECTEDPROOFvi Forewordmade up of a list of contributions related with these topics. All the papers have 25satisfied a strict refereeing process by anonymous specialists before publishing. 26It is a pleasure, for all of us, to dedicate this volume to her.AQ3 27Ankara, Turkey S¸engül Kuru 28Valladolid, Spain Javier Negro 29Valladolid, Spain Luis M. Nieto 30March 31, 2019 31UNCORRECTEDPROOFContents 1Curvature as an Integrable Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2Angel Ballesteros, Alfonso Blasco, and Francisco J. Herranz 3Trends in Supersymmetric Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4David J. Fernández C 5Coherent States in Quantum Optics: An Oriented Overview . . . . . . . . . . . . . . . 69 6Jean-Pierre Gazeau 7Higher Order Quantum Superintegrability: A New “Painlevé 8Conjecture” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 9Ian Marquette and Pavel Winternitz 10Supersymmetries in Schrödinger–Pauli Equations and in 11Schrödinger Equations with Position Dependent Mass . . . . . . . . . . . . . . . . . . . . . . 133 12A. G. Nikitin 13Nonlinear Supersymmetry as a Hidden Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 163 14Mikhail S. Plyushchay 15Coherent and Squeezed States: Introductory Review of Basic 16Notions, Properties, and Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 17Oscar Rosas-Ortiz 18Trace Formulas Applied to the Riemann ζ -Function . . . . . . . . . . . . . . . . . . . . . . . . 231 19Mark S. Ashbaugh, Fritz Gesztesy, Lotfi Hermi, Klaus Kirsten, Lance 20Littlejohn, and Hagop Tossounian 21Real- and Complex-Energy Non-conserving Particle Number 22Pairing Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 23Rodolfo M. Id Betan 24Jacobi Polynomials as su(2, 2) Unitary Irreducible Representation . . . . . . . 267 25Enrico Celeghini, Mariano A. del Olmo, and Miguel Angel Velasco 26viiUNCORRECTEDPROOFviii ContentsInfinite Square-Well, Trigonometric Pöschl-Teller and Other 27Potential Wells with a Moving Barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 28Alonso Contreras-Astorga and Véronique Hussin 29Variational Method Applied to Schrödinger-Like Equation . . . . . . . . . . . . . . . . 301 30Elso Drigo Filho, R. M. Ricotta, and N. F. Ribeiro 31The Lippmann–Schwinger Formula and One Dimensional Models 32with Dirac Delta Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 33Fatih Erman, Manuel Gadella, and Haydar Uncu 34Hermite Coherent States for Quadratic Refractive Index Optical Media . 323 35Zulema Gress and Sara Cruz y Cruz 36Analysis of CPN−1 Sigma Models via Soliton Surfaces . . . . . . . . . . . . . . . . . . . . . . 341 37P. P. Goldstein and A. M. Grundland 38On the Equivalence Between Type I Liouville Dynamical Systems 39in the Plane and the Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 40Miguel A. González León, Juan Mateos Guilarte, and M. de la Torre 41Mayado 42Construction of Partial Differential Equations with Conditional 43Symmetries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 44Decio Levi, Miguel A. Rodríguez, and Zora Thomova 45An Integro-Differential Equation of the Fractional Form: Cauchy 46Problem and Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 47Fernando Olivar-Romero and Oscar Rosas-Ortiz 48Quasi-Integrability and Some Aspects of SU(3) Toda Field Theory . . . . . . . 395 49Wojtek Zakrzewski 50On Some Aspects of Unitary Evolution Generated 51by Non-Hermitian Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 52Miloslav Znojil 53Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 54UNCORRECTEDPROOFContributors 1Mark S. Ashbaugh Department of Mathematics, University of Missouri, 2Columbia, MO, USA 3Ángel Ballesteros Departamento de Física, Facultad de Ciencias, Universidad de 4Burgos, Burgos, Spain 5Alfonso Blasco Sanz Departamento de Física, Escuela Politécnica Superior, 6Universidad de Burgos, Burgos, Spain 7Enrico Celeghini Dpto. di Fisica, Università di Firenze and INFN-Sezione di 8Firenze, Sesto Fiorentino, Firenze, Italy 9Alonso Contreras-Astorga Departamento de Física, Cinvestav, México, Mexico 10Department of Physics, Indiana University Northwest, Gary, IN, USA 11S. Cruz y Cruz Sara Instituto Politécnico Nacional, UPIITA, Ciudad de México, 12Mexico 13M. de la Torre Mayado Departamento de Física Fundamental and IUFFyM, 14Universidad de Salamanca, Salamanca, Spain 15Mariano A. del Olmo Departamento de Física Teórica, Atómica y Óptica, Univer- 16sidad de Valladolid, Valladolid, Spain 17Elso Drigo Filho Instituto de Biociências, Letras e Ciências Exatas, IBILCE- 18UNESP Rua Cristóvao Colombo, Sao José do Rio Preto, SP, Brazil 19Fatih Erman Department of Mathematics, Izmir Institute of Technology, Urla, 20Izmir, Turkey 21David J. Fernández C Departamento de Física, Cinvestav, Ciudad de México, 22México 23Manuel Gadella Departamento de Física Teórica, Atómica y Óptica, Universidad 24de Valladolid, Valladolid, Spain 25ixUNCORRECTEDPROOFx ContributorsJean-Pierre Gazeau APC, UMR 7164, Univ Paris Diderot, Sorbonne Paris Cité, 26Paris, France 27Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro, RJ, Brazil 28Fritz Gesztesy Department of Mathematics, Baylor University, Waco, TX, USA 29P. P. Goldstein Theoretical Physics Division, National Centre for Nuclear 30Research, Warsaw, Poland 31Miguel A. González León Departamento de Matemática Aplicada and IUFFyM, 32Universidad de Salamanca, Salamanca, Spain 33Zulema Gress Mendoza Universidad Autónoma del Estado de Hidalgo, Ciudad 34del Conocimiento, Hidalgo, México 35A. M. Grundland Centre de Recherches Mathématiques, Université de Montréal, 36Montréal, QC, Canada 37Département de Mathématiques et Informatique, Université du Québec à Trois- 38Rivières, Trois-Rivières, QC, Canada 39Juan Mateos Guilarte Departamento de Fìsica Fundamental and IUFFyM, Uni- 40versidad de Salamanca, Salamanca, Spain 41Lotfi Hermi Department of Mathematics and Statistics, Florida International 42University, Miami, FL, USA 43Francisco J. Herranz Departamento de Física, Escuela Politécnica Superior, 44Universidad de Burgos, Burgos, Spain 45Véronique Hussin Centre de Recherches Mathématique, Département de Mathé- 46matiques et de Statistique, Universitè de Montréal, Montréal, QC, Canada 47Rodolfo M. Id Betan, Instituto de Física Rosario (CONICET-UNR), Facultad de 48Ciencias Exactas, Ingeniería y Agrimensura (UNR), Santa Fe, Argentina 49Instituto de Estudios Nucleares y Radiaciones Ionizantes (UNR), Ocampo y Esmer- 50alda, Rosario, Santa Fe, Argentina 51Klaus Kirsten GCAP-CASPER, Department of Mathematics, Baylor University, 52Waco, TX, USA 53Decio Levi INFN, Sezione Roma Tre, Roma, Italy 54Lance Littlejohn Department of Mathematics, Baylor University, Waco, TX, USA 55Ian Marquette School of Mathematics and Physics, The University of Queens- 56land, Brisbane, St-Lucia, QLD, Australia 57A. G. Nikitin Institute of Mathematics, National Academy of Sciences of Ukraine, 58Kyiv, Ukraine 59Fernando Olivar-Romero Physics Department, Cinvestav, Mexico City, Mexico 60UNCORRECTEDPROOFContributors xiMikhail S. Plyushchay Departamento de Física, Universidad de Santiago de 61Chile, Santiago, Chile 62N. F. Ribeiro Centro Universitário do Norte Paulista, UNORP R. Ipiranga, Sao 63José do Rio Preto, SP, Brazil 64R.M. Ricotta Faculdade de Tecnologia de Sao Paulo, FATEC/SP-CEETEPS Praca 65Coronel Fernando Prestes, Sao Paulo, Brazil 66Miguel A. Rodríguez Departamento de Física Teórica, Universidad Complutense 67de Madrid, Madrid, Spain 68Oscar Rosas-Ortiz Physics Department, Cinvestav, Mexico City, Mexico 69Zora Thomova SUNY Polytechnic Institute, Utica, NY, USA 70Hagop Tossounian Department of Mathematics, Baylor University, Waco, TX, 71USA 72Haydar Uncu Department of Physics, Adnan Menderes University, Aydin, Turkey 73Miguel Angel Velasco Departamento de Física Teórica, Atómica y Óptica, Uni- 74versidad de Valladolid, Valladolid, Spain 75Pavel Winternitz Centre de recherches mathématiques et Département de Mathé- 76matiques et de Statistique, Université de Montréal, Montréal, QC,Canada 77Wojtek Zakrzewski Durham University, Durham, UK 78Miloslav Znojil NPI ASCR, Rˇež, Czech Republic 79AQ4UNCORRECTEDPROOFAUTHOR QUERIESAQ1. Please provide ISSN for Print and Electronic if applicable.AQ2. Please check if “6th International Workshop on” should be changed to “6thInternational Workshop on”.AQ3. Please check and confirm the presentation of signature details and correct ifnecessary.AQ4. Please check the edit made in author names “Mikhail S. Plyushchay, S. Cruzy Cruz, N. F. Ribeiro, M. de la Torre Mayado, and Oscar Rosas-Ortiz” in thecontributors and correct if necessary.

Integrability, Supersymmetry and Coherent States. A volume in honour of Professor Véronique Hussin

This volume shares and makes accessible new research lines and recent results in several branches of theoretical and mathematical physics, among them Quantum Optics, Coherent States, Integrable Systems, SUSY Quantum Mechanics, and Mathematical Methods in Physics. In addition to a selection of the contributions presented at the "6th International Workshop on New Challenges in Quantum Mechanics: Integrability and Supersymmetry", held in Valladolid, Spain, 27-30 June 2017, several high quality contributions from other authors are also included. The conference gathered 60 participants from many countries working in different fields of Theoretical Physics, and was dedicated to Prof. Véronique Hussin—an internationally recognized expert in many branches of Mathematical Physics who has been making remarkable contributions to this field since the 1980s. The reader will find interesting reviews on the main topics from internationally recognized experts in each field, as well as other original contributions, all of which deal with recent applications or discoveries in the aforementioned areas

Kuru, Şengül

Negro, Javier

Nieto, Luis

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Integrability, supersymmetry and coherent states: a volume in honour of Professor Véronique Hussin

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Integrability, Supersymmetry and Coherent States. A volume in honour of Professor Véronique Hussin

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