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Mark Vishik was one of the prominent figures in the theory of
partial differential equations. His ground-breaking contributions
were instrumental in integrating the methods of functional analysis
into this theory. In this book, friends and pupils of Mark Vishik
remember his life and work. From his early years as a student to
his connection with the Lwow school of Stephan Banach and to his
later career as a respected teacher and mentor, Vishik's legacy is
explored in detail. His research and pedagogical work influenced
hundreds of undergraduate and graduate students, many of whom went
on to become leading figures in their own right. The reader is
introduced to a number of remarkable scientists whose lives
intersected with Vishik's, including S. Banach, J. Schauder, I. N.
Vekua, N. I. Muskhelishvili, L. A. Lyusternik, I. G. Petrovskii, S.
L. Sobolev, I. M. Gelfand, M. G. Krein, A. N. Kolmogorov, N. I.
Akhiezer, J. Leray, J.-L. Lions, L. Schwartz, L. Nirenberg, and
others.
The focus of the present work is nonrelativistic and relativistic
quantum mechanics with standard applications to the hydrogen atom.
The author has aimed at presenting quantum mechanics in a
comprehensive yet accessible for mathematicians and other
non-physicists. The genesis of quantum mechanics, its applications
to basic quantum phenomena, and detailed explanations of the
corresponding mathematical methods are presented. The exposition is
formalized (whenever possible) on the basis of the coupled
Schroedinger, Dirac and Maxwell equations. Aimed at upper graduate
and graduate students in mathematical and physical science studies.
The focus of the present work is nonrelativistic and relativistic
quantum mechanics with standard applications to the hydrogen atom.
The author has aimed at presenting quantum mechanics in a
comprehensive yet accessible for mathematicians and other
non-physicists. The genesis of quantum mechanics, its applications
to basic quantum phenomena, and detailed explanations of the
corresponding mathematical methods are presented. The exposition is
formalized (whenever possible) on the basis of the coupled
Schroedinger, Dirac and Maxwell equations. Aimed at upper graduate
and graduate students in mathematical and physical science studies.
ThisbookisintendedtogivethereaderanopportunitytomastersolvingPDEpr-
lems.
Ourmaingoalwastohaveaconcisetextthatwouldcovertheclassicaltools
ofPDEtheorythatareusedintoday'sscienceandengineering, suchaschar-
teristics, thewavepropagation, theFouriermethod, distributions,
Sobolevspaces, fundamentalsolutions, andGreen'sfunctions.
WhileintroductoryFouriermethod
-basedPDEbooksdonotgiveanadequatedescriptionoftheseareas, themore
advancedPDEbooksarequitetheoreticalandrequireahighlevelofmathematical
backgroundfromareader. Thisbookwaswrittenspeci?callyto?llthisgap,
sat- fyingthedemandofthewiderangeofenduserswhoneedtheknowledgeofhow
tosolvethePDEproblemsandatthesametimearenotgoingtospecializeinthis
areaofmathematics. Arguably, thisistheshortestPDEcourse,
whichstretchesfar beyondcommon, Fouriermethod-basedPDEtexts.
Forexample, Hab03], which
isacommonthoroughtextbookonpartialdifferentialequations,
teachesasimilar setoftoolswhilebeingabout?vetimeslonger.
Thebookisproblem-oriented. Thetheoreticalpartisrigorousyetshort.
So-
timeswereferthereadertotextbooksthatgivewidercoverageofthetheory.
Yet, - portanttheoreticaldetailsarepresentedwithcare,
whilethehintsgivethereaderan
opportunitytorestoretheargumentstothefullrigor.
Manyexamplesfromphysics
areintendedtokeepthebookintuitiveforthereaderandtoillustratetheapplied
natureofthesubject.
Thebookwillbeusefulforanyhigher-levelundergraduatecourseandforse-
studyforbothgraduateandhigher-levelundergraduatestudents,
andforanys- cialtyinsciences.
ItsRussianversionhasbeenastandardproblem-solvingmanual
atMoscowStateUniversitysince1988, andisalsousedbystudentsofSt.
Pete- burgUniversityandNovosibirskUniversities.
ItsSpanishversionisusedatMorelia UniversityinMexico,
whiletheEnglishdrafthasalreadybeenusedinViennaU-
versityandatTexasA&MUniversity. Forfurtherreadingwerecommend
Str92], Eva98], and EKS99]. Mu]nchen, AlexanderKomech August2007
AndrewKomech v Acknowledgements
The?rstauthorisindebtedtoMargaritaKorotkinaforthefortunatesuggestionto
writethisbook, toA. F. Filippov, A. S. Kalashnikov, M. A. Shubin,
T. D. Ventzel, andM. I.
Vishikforcheckingthe?rstversionofthemanuscriptandfortheadvice.
BothauthorsaregratefultoH. Spohn(TechnischeUniversita]t,
Mu]nchen)andto E. Zeidler(Max-PlanckInstituteforMathematics,
Leipzig)fortheirhospitalityand supportduringtheworkonthebook.
BothauthorsweresupportedbyInstituteforInformationTransmissionProblems
(RussianAcademyofSciences).
The?rstauthorwassupportedbytheDepartment
ofMechanicsandMathematicsofMoscowStateUniversity, bytheAlexandervon
HumboldtResearchAward, FWFGrantP19138-N13, andtheGrantsofRFBR. The
secondauthorwassupportedbyTexasA&MUniversityandbytheNationalScience
FoundationunderGrantsDMS-0621257andDMS-0600863. vii Contents 1
Hyperbolicequations. Methodofcharacteristics. . . . . . . . . . . .
. . . . . . . 1 1 Derivationofthed'Alembertequation. . . . . . . .
. . . . . . . . . . . . . . . . . 1 2
Thed'Alembertmethodforin?nitestring . . . . . . . . . . . . . . . .
. . . . . . 7 3 Analysisofthed'Alembertformula. . . . . . . . . . .
. . . . . . . . . . . . . . . . 12 4
Second-orderhyperbolicequationsintheplane . . . . . . . . . . . . .
. . . . 19 5 Semi-in?nitestring. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 30 6 Finitestring . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 44 7 Waveequationwithmanyindependentvariables
. . . . . . . . . . . . . . . . 46 8 Generalhyperbolicequations. .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2
TheFouriermethod. . . ."
This book presents a new and original method for the solution of
boundary value problems in angles for second-order elliptic
equations with constant coefficients and arbitrary boundary
operators. This method turns out to be applicable to many different
areas of mathematical physics, in particular to diffraction
problems in angles and to the study of trapped modes on a sloping
beach. Giving the reader the opportunity to master the techniques
of the modern theory of diffraction, the book introduces methods of
distributions, complex Fourier transforms, pseudo-differential
operators, Riemann surfaces, automorphic functions, and the
Riemann-Hilbert problem. The book will be useful for students,
postgraduates and specialists interested in the application of
modern mathematics to wave propagation and diffraction problems.
This book gives a concise introduction to Quantum Mechanics with a
systematic, coherent, and in-depth explanation of related
mathematical methods from the scattering theory and the theory of
Partial Differential Equations.The book is aimed at graduate and
advanced undergraduate students in mathematics, physics, and
chemistry, as well as at the readers specializing in quantum
mechanics, theoretical physics and quantum chemistry, and
applications to solid state physics, optics, superconductivity, and
quantum and high-frequency electronic devices.The book utilizes
elementary mathematical derivations. The presentation assumes only
basic knowledge of the origin of Hamiltonian mechanics, Maxwell
equations, calculus, Ordinary Differential Equations and basic
PDEs. Key topics include the Schroedinger, Pauli, and Dirac
equations, the corresponding conservation laws, spin, the hydrogen
spectrum, and the Zeeman effect, scattering of light and particles,
photoelectric effect, electron diffraction, and relations of
quantum postulates with attractors of nonlinear Hamiltonian PDEs.
Featuring problem sets and accompanied by extensive contemporary
and historical references, this book could be used for the course
on Quantum Mechanics and is also suitable for individual study.
ThisbookisintendedtogivethereaderanopportunitytomastersolvingPDEpr-
lems.
Ourmaingoalwastohaveaconcisetextthatwouldcovertheclassicaltools
ofPDEtheorythatareusedintoday'sscienceandengineering, suchaschar-
teristics, thewavepropagation, theFouriermethod, distributions,
Sobolevspaces, fundamentalsolutions, andGreen'sfunctions.
WhileintroductoryFouriermethod
-basedPDEbooksdonotgiveanadequatedescriptionoftheseareas, themore
advancedPDEbooksarequitetheoreticalandrequireahighlevelofmathematical
backgroundfromareader. Thisbookwaswrittenspeci?callyto?llthisgap,
sat- fyingthedemandofthewiderangeofenduserswhoneedtheknowledgeofhow
tosolvethePDEproblemsandatthesametimearenotgoingtospecializeinthis
areaofmathematics. Arguably, thisistheshortestPDEcourse,
whichstretchesfar beyondcommon, Fouriermethod-basedPDEtexts.
Forexample, Hab03], which
isacommonthoroughtextbookonpartialdifferentialequations,
teachesasimilar setoftoolswhilebeingabout?vetimeslonger.
Thebookisproblem-oriented. Thetheoreticalpartisrigorousyetshort.
So-
timeswereferthereadertotextbooksthatgivewidercoverageofthetheory.
Yet, - portanttheoreticaldetailsarepresentedwithcare,
whilethehintsgivethereaderan
opportunitytorestoretheargumentstothefullrigor.
Manyexamplesfromphysics
areintendedtokeepthebookintuitiveforthereaderandtoillustratetheapplied
natureofthesubject.
Thebookwillbeusefulforanyhigher-levelundergraduatecourseandforse-
studyforbothgraduateandhigher-levelundergraduatestudents,
andforanys- cialtyinsciences.
ItsRussianversionhasbeenastandardproblem-solvingmanual
atMoscowStateUniversitysince1988, andisalsousedbystudentsofSt.
Pete- burgUniversityandNovosibirskUniversities.
ItsSpanishversionisusedatMorelia UniversityinMexico,
whiletheEnglishdrafthasalreadybeenusedinViennaU-
versityandatTexasA&MUniversity. Forfurtherreadingwerecommend
Str92], Eva98], and EKS99]. Mu]nchen, AlexanderKomech August2007
AndrewKomech v Acknowledgements
The?rstauthorisindebtedtoMargaritaKorotkinaforthefortunatesuggestionto
writethisbook, toA. F. Filippov, A. S. Kalashnikov, M. A. Shubin,
T. D. Ventzel, andM. I.
Vishikforcheckingthe?rstversionofthemanuscriptandfortheadvice.
BothauthorsaregratefultoH. Spohn(TechnischeUniversita]t,
Mu]nchen)andto E. Zeidler(Max-PlanckInstituteforMathematics,
Leipzig)fortheirhospitalityand supportduringtheworkonthebook.
BothauthorsweresupportedbyInstituteforInformationTransmissionProblems
(RussianAcademyofSciences).
The?rstauthorwassupportedbytheDepartment
ofMechanicsandMathematicsofMoscowStateUniversity, bytheAlexandervon
HumboldtResearchAward, FWFGrantP19138-N13, andtheGrantsofRFBR. The
secondauthorwassupportedbyTexasA&MUniversityandbytheNationalScience
FoundationunderGrantsDMS-0621257andDMS-0600863. vii Contents 1
Hyperbolicequations. Methodofcharacteristics. . . . . . . . . . . .
. . . . . . . 1 1 Derivationofthed'Alembertequation. . . . . . . .
. . . . . . . . . . . . . . . . . 1 2
Thed'Alembertmethodforin?nitestring . . . . . . . . . . . . . . . .
. . . . . . 7 3 Analysisofthed'Alembertformula. . . . . . . . . . .
. . . . . . . . . . . . . . . . 12 4
Second-orderhyperbolicequationsintheplane . . . . . . . . . . . . .
. . . . 19 5 Semi-in?nitestring. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 30 6 Finitestring . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 44 7 Waveequationwithmanyindependentvariables
. . . . . . . . . . . . . . . . 46 8 Generalhyperbolicequations. .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2
TheFouriermethod. . . ."
This monograph is the first to present the theory of global
attractors of Hamiltonian partial differential equations. A
particular focus is placed on the results obtained in the last
three decades, with chapters on the global attraction to stationary
states, to solitons, and to stationary orbits. The text includes
many physically relevant examples and will be of interest to
graduate students and researchers in both mathematics and physics.
The proofs involve novel applications of methods of harmonic
analysis, including Tauberian theorems, Titchmarsh's convolution
theorem, and the theory of quasimeasures. As well as the underlying
theory, the authors discuss the results of numerical simulations
and formulate open problems to prompt further research.
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