Table Of ContentSpringer Proceedings in Physics 184
Fabio Bagarello
Roberto Passante
Camillo Trapani E ditors
Non-Hermitian
Hamiltonians
in Quantum
Physics
Selected Contributions from the 15th
International Conference on Non-Hermitian
Hamiltonians in Quantum Physics, Palermo,
Italy, 18–23 May 2015
Springer Proceedings in Physics
Volume 184
The series Springer Proceedings in Physics, founded in 1984, is devoted to timely
reports of state-of-the-art developments in physics and related sciences. Typically
based on material presented at conferences, workshops and similar scientific
meetings, volumes published in this series will constitute a comprehensive
up-to-date source of reference on a field or subfield of relevance in contemporary
physics. Proposals must include the following:
– name, place and date of the scientific meeting
– a link to the committees (local organization, international advisors etc.)
– scientific description of the meeting
– list of invited/plenary speakers
– an estimate of the planned proceedings book parameters (number of pages/
articles, requested number of bulk copies, submission deadline).
More information about this series at http://www.springer.com/series/361
Fabio Bagarello Roberto Passante
(cid:129)
Camillo Trapani
Editors
Non-Hermitian Hamiltonians
in Quantum Physics
Selected Contributions from the 15th
International Conference on Non-Hermitian
Hamiltonians in Quantum Physics, Palermo,
–
Italy, 18 23 May 2015
123
Editors
FabioBagarello Camillo Trapani
Dipartimento di Energia Dipartimento di Matematica eInformatica
Universitàdegli Studi diPalermo Universitàdegli Studi diPalermo
Palermo Palermo
Italy Italy
RobertoPassante
Dipartimento di Fisica eChimica
Universitàdegli Studi diPalermo
Palermo
Italy
ISSN 0930-8989 ISSN 1867-4941 (electronic)
SpringerProceedings in Physics
ISBN978-3-319-31354-2 ISBN978-3-319-31356-6 (eBook)
DOI 10.1007/978-3-319-31356-6
LibraryofCongressControlNumber:2016936644
©SpringerInternationalPublishingSwitzerland2016
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the 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
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
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
authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor
foranyerrorsoromissionsthatmayhavebeenmade.
Printedonacid-freepaper
ThisSpringerimprintispublishedbySpringerNature
TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland
Preface
This volume collects the selected contributions presented at or inspired by the
15th International Workshop on Pseuso-Hermitian Hamiltonians in Quantum
Physics (PHHQP15), held in Palermo, Italy, from May 18 to 23, 2015. This
workshop was the 15th in the series of international meetings that was started in
2003. These meetings were mainly attended by mathematicians and physicists
interested in the study of non-Hermitian operators and Hamiltonians, and in their
physical applications. About 80 mathematicians and physicists attended the 2015
Workshop in Palermo.
Eventhoughmathematicianshavedeeplystudiedseveralaspectsofthespectral
theoryofoperatorssincelongtime,therealizationthatnon-HermitianHamiltonians
with PT symmetry may have a real spectrum has produced a growing interest in
theoretical physicists for this subject. From the mathematical side this renewed
perspective concerning operators with real spectrum has put on the stage new
methods aimed to find conditions for a non-self-adjoint operator to have a real
spectrum or it has led to revisiting (and, often, generalizing) older concepts (sim-
ilarity, affinity, metric operators, etc.) as tools for studying this problem. From a
physical point of view the main outcome of this unconventional approach to
quantummechanicshasbeentheexplorationofseveralnewandinterestingmodels.
Started as a pure mathematical problem, the subject of non-Hermitian
Hamiltonians with PT (parity-time) symmetry has rapidly grown in the past
years. It has also attracted much interest for its possible applications in physics,
since when it was shown that non-Hermitian Hamiltonians with PT symmetry can
have a real spectrum.
Nowadays, in fact, PT-symmetric non-Hermitian Hamiltonians have found
application in several areas of physics, for example in quantum optics, condensed
matter physics, non-equilibrium statistical physics, and quantum field theory, from
boththetheoreticalandexperimentalpointsofview.Typicalimportantsystemsthat
can be described by non-Hermitian Hamiltonians endowed with PT symmetry are
open systems with balanced gain–loss terms, where gain–loss mechanisms break
theHermiticitywhilepreservingthePTsymmetry.Realisticexamplesaregivenby
v
vi Preface
opticalwaveguidesandperiodiclattices with balancedabsorptionoramplification.
Otherrelevantaspectsthathavereceivedgreatattentioninrecenttimesare,among
the others, PT-symmetry breaking phase transitions and formation of exceptional
points and spectral singularities.
The papers in this volume will cover several aspects of PT-symmetric
non-Hermitian Hamiltonians, investigating both mathematical and physical
aspects of the research topics mentioned above.
Palermo, Italy Fabio Bagarello
Roberto Passante
Camillo Trapani
Acknowledgments
The organization of the 15th International Workshop on Pseuso-Hermitian
Hamiltonians in Quantum Physics was financially supported by
The President of the Assemblea Regionale Siciliana
The European Physical Society
The Gruppo Nazionale per l’ Analisi Matematica, la Probabilità e le loro
Applicazioni and the Gruppo Nazionale per la Fisica Matematica of the Istituto
Nazionale di Alta Matematica “F. Severi”
Università di Palermo
Dipartimento di Matematica e Informatica, Università di Palermo
Dipartimento di Energia, Ingegneria dell’Informazione e Modelli Matematici,
Università di Palermo
Dipartimento di Fisica e Chimica, Università di Palermo
Banca Nuova
and sponsored by the Società Italiana di Fisica.
We thank these Institutions for making possible this meeting. We also thank
speakers, contributors, chairpersons, and participants.
Wealsoacknowledgethesupportof3PERIODICOs.n.c.SocietàdiIngegneria
and Bidditti—delicious Italian food.
Our special thanks go to the local organizing committee, Giorgia Bellomonte,
FrancescoGargano,Margherita Lattuca,SalvatoreSpagnolo,SalvatoreTriolo,and
Francesco Tschinke, for the great job they did.
Palermo, Italy Fabio Bagarello
Roberto Passante
Camillo Trapani
vii
Contents
Real Discrete Spectrum of Complex PT-Symmetric Scattering
Potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Zafar Ahmed, Joseph Amal Nathan, Dhruv Sharma and Dona Ghosh
Geometrical and Asymptotical Properties of Non-Selfadjoint
Induction Equation with the Jump of the Velocity Field. Time
Evolution and Spatial Structure of the Magnetic Field . . . . . . . . . . . . . 11
Anna I. Allilueva and Andrei I. Shafarevich
PT Symmetric Classical and Quantum Cosmology. . . . . . . . . . . . . . . . 29
Alexander A. Andrianov, Chen Lan and Oleg O. Novikov
Operator (Quasi-)Similarity, Quasi-Hermitian Operators
and All that . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Jean-Pierre Antoine and Camillo Trapani
Generalized Jaynes-Cummings Model with a Pseudo-Hermitian:
A Path Integral Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Mekki Aouachria
Exceptional Points in a Non-Hermitian Extension
of the Jaynes-Cummings Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 83
Fabio Bagarello, Francesco Gargano, Margherita Lattuca,
Roberto Passante, Lucia Rizzuto and Salvatore Spagnolo
D(cid:1)Deformed and SUSY-Deformed Graphene: First Results. . . . . . . . . 97
F. Bagarello and M. Gianfreda
Localised Nonlinear Modes in the PT-Symmetric Double-Delta Well
Gross-Pitaevskii Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
I.V. Barashenkov and D.A. Zezyulin
Exactly Solvable Wadati Potentials in the PT-Symmetric
Gross-Pitaevskii Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
I.V. Barashenkov, D.A. Zezyulin and V.V. Konotop
ix
x Contents
The EMM and the Spectral Analysis of a Non Self-adjoint
Hamiltonian on an Infinite Dimensional Hilbert Space . . . . . . . . . . . . . 157
Natalia Bebiano and João da Providência
Bessel Sequences, Riesz-Like Bases and Operators in Triplets of
Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Giorgia Bellomonte
Geometric Aspects of Space-Time Reflection Symmetry in Quantum
Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Carl M. Bender, Dorje C. Brody, Lane P. Hughston
and Bernhard K. Meister
MathematicalandPhysicalMeaningoftheCrossingsofEnergyLevels
in PT-Symmetric Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Denis I. Borisov and Miloslav Znojil
Non-unitary Evolution of Quantum Logics. . . . . . . . . . . . . . . . . . . . . . 219
Sebastian Fortin, Federico Holik and Leonardo Vanni
A Unifying E2-Quasi Exactly Solvable Model. . . . . . . . . . . . . . . . . . . . 235
Andreas Fring
Sublattice Signatures of Transitions in a PT-Symmetric Dimer
Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
Andrew K. Harter and Yogesh N. Joglekar
Physical Aspect of Exceptional Point in the Liouvillian Dynamics
for a Quantum Lorentz Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Kazunari Hashimoto, Kazuki Kanki, Satoshi Tanaka and Tomio Petrosky
Some Features of Exceptional Points . . . . . . . . . . . . . . . . . . . . . . . . . . 281
W.D. Heiss
Spontaneous Breakdown of a PT-Symmetry in the Liouvillian
Dynamics at a Non-Hermitian Degeneracy Point . . . . . . . . . . . . . . . . . 289
Kazuki Kanki, Kazunari Hashimoto, Tomio Petrosky and Satoshi Tanaka
Pseudo-Hermitian b-Ensembles with Complex Eigenvalues. . . . . . . . . . 305
Gabriel Marinello and Mauricio Porto Pato
Green’s Function of a General PT-Symmetric Non-Hermitian
Non-central Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
Brijesh Kumar Mourya and Bhabani Prasad Mandal
Non-Hermitian Quantum Annealing and Superradiance. . . . . . . . . . . . 329
Alexander I. Nesterov, Gennady P. Berman, Fermín Aceves de la Cruz
and Juan Carlos Beas Zepeda
