Advances in quantum mechanics : contemporary trends and open problems

292 Pages·2017·3.772 MB·English

by  Dell'Antonio, Gianfausto, Michelangeli, Alessandro

Springer INdAM Series 18
Alessandro Michelangeli
Gianfausto Dell'Antonio E ditors
Advances in
Quantum
Mechanics
Contemporary Trends and Open
Problems
Springer INdAM Series
Volume 18
Editor-in-Chief
G.Patrizio
SeriesEditors
C.Canuto
G.Coletti
G.Gentili
A.Malchiodi
P.Marcellini
E.Mezzetti
G.Moscariello
T.Ruggeri
Moreinformationaboutthisseriesathttp://www.springer.com/series/10283
Alessandro Michelangeli • Gianfausto Dell’Antonio
Editors
Advances in Quantum
Mechanics
Contemporary Trends and Open Problems
123
Editors
AlessandroMichelangeli GianfaustoDell’Antonio
InternationalSch.forAdvancedStudies InternationalSch.forAdvancedStudies
SISSA SISSAandSapienzaUniversityofRome
Trieste,Italy Trieste,Italy
ISSN2281-518X ISSN2281-5198 (electronic)
SpringerINdAMSeries
ISBN978-3-319-58903-9 ISBN978-3-319-58904-6 (eBook)
DOI10.1007/978-3-319-58904-6
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Preface
This volume is a collection of recent contributions and up-to-date surveys on
many contemporary trends in the mathematics of quantum mechanics, and more
generally on mathematical problems arising in quantum many-body dynamics,
quantumgraphtheory,coldatomsandunitarygases.Specialemphasisisdevotedto
developmentof the specific mathematical tools needed, including linear and non-
linear Schrödinger equations, topological invariants, non-commutative geometry,
resonancesandoperatorextensiontheory.
Most of the contributorsare leading international experts or recognised young
researchersinmathematicalphysics,PDEtheoryandoperatortheory.Thematerial
thattheypresentisthefruitofrecentstudiesthathavealreadybecomeareference
inthecommunity.Theunderlyingmotivationfromcondensedmatterphysics,solid
statephysicsandultra-coldatomphysics,andthetopicalityoftheresearchtopics,
givethevolumeadistinctiveperspectiveattheedgeofmathematicsandphysics.
A large part of the material was presented and discussed thoroughly on the
occasionoftheINdAMinternationalmeetingentitled“ContemporaryTrendsinthe
MathematicsofQuantumMechanics”,whichtookplaceinRomefrom4to8July
2016andwhichwehadthehonouroforganisingthankstoaverygenerousfunding
andmosthelpfullogisticsupportfromINdAM.Theremainderofthematerialwas
producedasafollow-uptothatmeetingorascloselyrelatedwork.
First and foremost, our thanks go to the scientific board of INdAM and the
responsibleadministrativestaffattheINdAMheadquartersinRomeforproviding
sucha stimulatingatmosphereandallthenecessarypracticalhelp.We wouldalso
like to warmly acknowledge all contributors and anonymous reviewers for their
careful work and the quality of their reports. Finally, we extend our gratitude to
the extremely supportive team of the INdAM Springer Series for their services
throughouttheeditingandpublishingprocess.
Trieste,Italy GianfaustoDell’Antonio
Trieste,Italy AlessandroMichelangeli
April2017
v
Contents
ShellInteractionsforDiracOperators ........................................ 1
NaiaraArrizabalaga
CorrelationInequalitiesforClassicalandQuantumXYModels........... 15
CostanzaBenassi,BenjaminLees,andDanielUeltschi
DissipativelyGeneratedEntanglement........................................ 33
FabioBenatti
AbelianGaugePotentialsonCubicLattices.................................. 47
M.Burrello,L.Lepori,S.Paganelli,andA.Trombettoni
Relative-Zeta and Casimir Energy for a Semitransparent
HyperplaneSelectingTransverseModes...................................... 71
ClaudioCacciapuoti,DavideFermi,andAndreaPosilicano
AnalysisofFluctuationsAroundNon-linearEffectiveDynamics .......... 99
SerenaCenatiempo
LogarithmicSobolevInequalitiesforanIdealBoseGas .................... 121
FabioCipriani
SphericalSchrödingerHamiltonians:SpectralAnalysis
andTimeDecay .................................................................. 135
LucaFanelli
OntheGroundStatefortheNLSEquationonaGeneralGraph.......... 153
DomenicoFinco
Self-AdjointExtensionsofDiracOperatorwithCoulombPotential....... 169
MatteoGallone
Dispersive Estimates for Schrödinger Operators with Point
InteractionsinR3 ................................................................ 187
FeliceIandoliandRaffaeleScandone
vii
viii Contents
ChernandFu–Kane–MeleInvariantsasTopologicalObstructions........ 201
DomenicoMonaco
NormApproximationforMany-BodyQuantumDynamics
andBogoliubovTheory.......................................................... 223
PhanThànhNamandMarcinNapiórkowski
EffectiveNon-linearDynamicsofBinaryCondensates
andOpenProblems.............................................................. 239
AlessandroOlgiati
Remarks on the Derivation of Gross-Pitaevskii Equation
withMagneticLaplacian........................................................ 257
AlessandroOlgiati
OntheInverseSpectralProblemsforQuantumGraphs.................... 267
M.OlivieriandD.Finco
Double-Barrier Resonances and Time Decay of the Survival
Probability:AToyModel ....................................................... 283
AndreaSacchetti
About the Editors
Prof. Gianfausto Dell’Antonio’s research focuses on axiomatic quantum field
theory,localfieldtheory,mathematicsofquantummechanics,criticalpointtheory,
stochasticprocesses,singularinteractions,andmany-bodyproblems.Hegraduated
intheoreticalphysicsinMilan,wasresearchassociateinCopenhagen(NielsBohr
institute),Zurich(ETH),andEvanson(Northwestern),thenprofessoroftheoretical
physicsinNaplesandprofessorofrationalmechanicsandmathematicalphysicsat
La Sapienza Rome. He held visiting professorships at the IHES Paris, Courant
Institute NY, The University of Marseille Luminy, Bielefeld University (as a
recipientofavonHumboldtprize),CERN,SISSATrieste,andtheInterdisciplinary
LaboratoryoftheAccademiadeiLincei.HeheldalsovisitingpositionsattheIAS
Princeton,EcolePolitechniqueParis, ParisDauphine,HarvardUniversity,andthe
MaxPlanckInstituteinMunich.
Dr. Alessandro Michelangeli’s research is in the field at the interface between
mathematical physics, functional analysis and non-linear dispersive PDE, and
operator theory, with a special focus on the mathematical methods for quantum
mechanical and condensed matter systems. He graduated in theoretical physics
in Pisa and in mathematical physics at SISSA Trieste, held faculty positions at
the LMU Munich and SISSA Trieste, and visiting positions at the University of
Cambridge,SISSATrieste,andBilkent.
ix
Shell Interactions for Dirac Operators
NaiaraArrizabalaga
Abstract In this notes we gather the latest results on spectral theory for the
couplingHCV,whereH D(cid:2)i˛(cid:3)rCmˇisthefreeDiracoperatorinR3,m>0
and V is a measure-valuedpotential.The potentialsunderconsiderationare given
in terms of surface measures on the boundaryof boundedregular domainsin R3.
We give three main results. We study the self-adjointness.We give a criterion for
the existence of point spectrum, with applications to electrostatic shell potentials,
V(cid:2), which dependon a parameter(cid:2) 2 R. Finally, we provean isoperimetric-type
inequalityfortheadmissiblerangeof(cid:2)’sforwhichthecouplingHCV(cid:2)generates
purepointspectrumin.(cid:2)m;m/.Theballistheuniqueoptimizerofthisinequality.
Keywords Dirac operator (cid:129) Self-adjointextension (cid:129) Shellinteraction (cid:129) Singular
integral
1 Introduction andMain Results
The quantum mechanical model presented in these notes is a shell interaction for
Dirac operators, which is nothing else than the free Dirac operator in R3 coupled
withameasure-valuedpotential.
Givenm (cid:4) 0, the freeDirac operatorin R3 is definedbyH D (cid:2)i˛ (cid:3)r Cmˇ;
where˛ D.˛1;˛2;˛3/,
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
˛j D (cid:3)0j (cid:3)0j forjD1;2;3; ˇ D I02 (cid:2)0I2 ; I2 D 1001 ;
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (1)
01 0(cid:2)i 1 0
and (cid:3)1 D 10 ; (cid:3)2 D i 0 ; (cid:3)3 D 0 (cid:2)1
N.Arrizabalaga((cid:2))
UniversityoftheBasqueCountry,UPV/EHU Apdo.644,48080Bilbao,Spain
e-mail:[email protected]
©SpringerInternationalPublishingAG2017 1
A.Michelangeli,G.Dell’Antonio(eds.),AdvancesinQuantumMechanics,
SpringerINdAMSeries18,DOI10.1007/978-3-319-58904-6_1