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Semiclassics, adiabatic decoupling and perturbed periodic structures [Elektronische Ressource] / vorgelegt von Eric Sträng

129 pages
Institut für Theoretische PhysikDirektor: Prof. Dr. Frank SteinerSEMICLASSICS, ADIABATIC DECOUPLINGAND PERTURBED PERIODICSTRUCTURESDISSERTATIONzur Erlangung des Doktorgrades Dr. rer. nat.der Fakultät für Naturwissenschaftender Universität Ulmvorgelegt vonEric Strängaus Lund.2008iAmtierendenDekan: Prof. Dr. PeterBäuerleErstgutachter: Prof. Dr. JensBolteZweitgutachter: Prof. Dr. FrankSteinerTagderPromotion: 9. Februar2009iiPrefaceThisthesisisanattempttoreporttheresearchIhaveconductedduringthethreeyears Ihave been active at the Institute of Theoretical Physicsof the universityinUlm.Ithasreallybeenaprivilegetoworkattheinstituteoftheoretical physics. IwouldthereforeliketothankallmembersandformermembersoftheinstituteIhavehadthepleasureofmeeting. Inparticular,IwouldliketothankJensBoltefor his support and supervision the last three years and Frank Steiner for thefriendly atmosphere he has created at the institute. I would also like to thankStefanKeppelerforrecommendingmetothelattertwo.Ihavehadthepleasuretoengageinmanyaninterestingdiscussionatthede-partment. IwouldliketocommendSebastianEndres,KonstantinGlaum,Hol-ger Janzer, Maximilian Köpke, Raphael Lamon, Sven Lustig, Benjamin Obert,Wolfgang Peter, Stefan Poppe, Nils Bezares–Roder, Hans–Michael Stiepan, Al-viseVerso, Daniel Waltner for their active participation.
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Institut für Theoretische Physik
Direktor: Prof. Dr. Frank Steiner
SEMICLASSICS, ADIABATIC DECOUPLING
AND PERTURBED PERIODIC
STRUCTURES
DISSERTATION
zur Erlangung des Doktorgrades Dr. rer. nat.
der Fakultät für Naturwissenschaften
der Universität Ulm
vorgelegt von
Eric Sträng
aus Lund.
2008i
AmtierendenDekan: Prof. Dr. PeterBäuerle
Erstgutachter: Prof. Dr. JensBolte
Zweitgutachter: Prof. Dr. FrankSteiner
TagderPromotion: 9. Februar2009iiPreface
ThisthesisisanattempttoreporttheresearchIhaveconductedduringthethree
years Ihave been active at the Institute of Theoretical Physicsof the university
inUlm.
Ithasreallybeenaprivilegetoworkattheinstituteoftheoretical physics. I
wouldthereforeliketothankallmembersandformermembersoftheinstituteI
havehadthepleasureofmeeting. Inparticular,IwouldliketothankJensBolte
for his support and supervision the last three years and Frank Steiner for the
friendly atmosphere he has created at the institute. I would also like to thank
StefanKeppelerforrecommendingmetothelattertwo.
Ihavehadthepleasuretoengageinmanyaninterestingdiscussionatthede-
partment. IwouldliketocommendSebastianEndres,KonstantinGlaum,Hol-
ger Janzer, Maximilian Köpke, Raphael Lamon, Sven Lustig, Benjamin Obert,
Wolfgang Peter, Stefan Poppe, Nils Bezares–Roder, Hans–Michael Stiepan, Al-
viseVerso, Daniel Waltner for their active participation. Iam particular thank-
fultoRaphaelwhohasintroduced metoquantumloopgravity.
Iamalsoindebtedtoallthepersonswhohavecontributedtotheproofread-
ingof this thesis. Thoseinclude Nils Bezares-Roder,SvenLustig, Sebastian En-
dres, Frank Steiner, Raphael Lamon and Jens Bolte. The largest part of the
burdenhasbeencarriedbyJens.
Iwould liketothankthegraduateschool"Analysis ofcomplexity, informa-
tion and evolution" of the Land Baden-Württembergfor the stipend which has
permitted me to do this doctoral thesis. I would furthermore like to thank the
members and professors of the graduate school. Amongst them I would par-
ticularly like to thank Wolfgang Arendt and Werner Balser of the Institute of
Applied Analysiswhohavealwaysbeenmorethanwillingtoassistme.
Last but not least, I would like to thank Anna, Axel and my parents whose
contributions to this thesis, though not of scientific nature, have been highly
appreciated.
12Abstract
This thesis deals with the consequences of periodic structures in quantum me-
chanics. This is done first in the usual semiclassical, i.e., relating properties of
classical mechanicsandquantummechanics. Inasecondstepwestudyperiodic
Schrödinger operators.
Thefirstchapterintroducesthemaintoolfordealingwithdifferentialequa-
tionswithperiodiccoefficients, namely,Floquettheory. Asanexampleitisap-
plied on the Hill equation. Hill equations appear in a class of spectral problem
forcertainSchrödingeroperatorswithperiodicpotentials. Asaspecialcase,the
solutions tothe Mathieu equation are then studied exemplifying the properties
of the corresponding spectrum. Special attention is brought to the properties
of the spectrum and the corresponding Floquet exponent of the corresponding
solution. ThisrelationisgivenbytheWhittakerformulawhichisexplicitupto
atermwhichcanbecharacterizedasthedeterminantofaninfinitedimensional
matrix. We present new results concerning this determinant. It is expressed as
the limiting case of a third order recurrence to which we can write any term
explicitly. Todothiswepresentanovelproceduretorepresentanylinearrecur-
rencewithoutconstantterms.
After ashortreviewofclassical mechanics, weintroduce quantum mechan-
ics in the context of the Weyl quantization. In effect this gives us a bridge
between classical dynamics and quantum mechanics. In particular, we intro-
duce semiclassical propagation results which, up to some known errors, de-
scribesthepropagationofaninitialGaußianunderquantumevolution [CR97].
ThisapproximationbreaksdownatwhatiscommonlyknownastheEhrenfest
time. We use these results to investigate the localization properties of an ini-
tial Gaußian up to the Ehrenfest time. We can show localization of the state
up to the Ehrenfest time under the assumption that the flow differential of the
correspondingclassicalmotionadmitsFloquetsolutionswithpurelyimaginary
Floquet exponents. In this case, we also show the existence of what we call
classical revivals which occur at multiples of the classical period. In the case of
periodic classical motion, the classical period is nothing else but the period of
themotion.
34
ThelastpartisastudyofthespectralpropertiesofperturbedperiodicSchrö-
dinger operators. This is also done in the context of semiclassics, however, the
perturbation parameter is now the strength of the perturbationǫ. To do this
we must consider operator valued Weyl calculus. We do this in the context of
τ−equivariantsymbols [PST03]. Asthe structureof the band spectrum of the
unperturbed operator remains unchanged under small enough perturbations,
theeffectivedynamicscanbedescribedadiabaticallyundercertainprerequisites.
We consider the construction of multi-valued WKB ansätze [LF91, EW96].
2Our results include Bohr-Sommerfeld quantization conditions moduloO ǫ .
Those are used to investigate the Wannier–Stark system. Such systems have
been investigated numerically showing that its evolution is almost dispersion-
less when restricted to one band [HKKM04, WKKM04]. We show that this
is due to the fact that the corresponding Berry phase of the WKB ansatz does
notdependontheenergy,suchthat,thespectrumofaparticlerestrictedtoone
2bandisthatofaharmonicoscillatormoduloO ǫ .Contents
1 Introduction 7
2 TheHill equation 11
2.1 Initialvalueproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Fundamentalsolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Floquettheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 TheHillequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.1 Thefundamentalsystem . . . . . . . . . . . . . . . . . . . . . 16
2.4.2 Hill’sdeterminantal equation . . . . . . . . . . . . . . . . . . 17
2.4.3 TheWhittaker-Hill formula . . . . . . . . . . . . . . . . . . 19
2.4.4 AnapproximationschemeforΔ(0,λ) . . . . . . . . . . . . 20
2.4.5 Asolutiontolinearrecursionswithoutconstantterms . 23
2.4.6 TheWhittaker-Formulainaclosedform . . . . . . . . . . 28
2.5 AsymptoticbehaviorandLyapunovstability . . . . . . . . . . . . 29
3 Classical mechanics 31
3.1 Structuresofclassicalmechanics . . . . . . . . . . . . . . . . . . . . . 31
3.2 Stabilityoftrajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Hamilton-Jacobitheory . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 Periodicstructuresofintegrability . . . . . . . . . . . . . . . . . . . . 35
4 Quantummechanics 37
4.1 Pseudo–differentialoperators . . . . . . . . . . . . . . . . . . . . . . . 38
4.2 The”classical“ insemiclassical . . . . . . . . . . . . . . . . . . . . . . 42
4.3 Constructionofapproximateevolution . . . . . . . . . . . . . . . . 44
4.3.1 Gaußianwavepackets . . . . . . . . . . . . . . . . . . . . . . 44
4.3.2 Aquadraticapproximationscheme . . . . . . . . . . . . . . 44
4.3.3 Semiclassical errors . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.4.1 Revivals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.4.2 Auniformboundfortheflowdifferential. . . . . . . . . . 52
56 CONTENTS
4.4.3 Thecaseofanon-diagonalizable Floquetmatrix . . . . . 53
4.5 TheEhrenfesttime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.6 Bohr–Sommerfeldquantizationconditions . . . . . . . . . . . . . . 57
4.7 WKB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.8 EBKquantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5 Perturbation of periodic operators 67
5.1 PeriodicSchrödingeroperators . . . . . . . . . . . . . . . . . . . . . . 68
5.1.1 TheBloch–Floquettransform . . . . . . . . . . . . . . . . . 68
5.1.2 Anequivalentapproach . . . . . . . . . . . . . . . . . . . . . 71
5.1.3 τ−equivariantsymbols . . . . . . . . . . . . . . . . . . . . . 72
5.2 PerturbedperiodicSchrödingeroperators . . . . . . . . . . . . . . . 73
5.2.1 Multi–scaleapproach . . . . . . . . . . . . . . . . . . . . . . . 73
5.2.2 Thealmostprojector . . . . . . . . . . . . . . . . . . . . . . . 75
5.3 Spectralasymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.3.1 ConstructionofWKBquasi–modes . . . . . . . . . . . . . 77
5.3.2 Quantization ofperturbedperiodicsystems . . . . . . . . 83
5.3.3 Generalstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.3.4 Anoteondegenerate eigenvalues . . . . . . . . . . . . . . . 86
5.4 TheWannier–StarkSystem . . . . . . . . . . . . . . . . . . . . . . . . 88
5.4.1 Spectralproperties . . . . . . . . . . . . . . . . . . . . . . . . 90
5.4.2 TheWannier–Starkladder . . . . . . . . . . . . . . . . . . . 93
5.4.3 Adiabatic propagationoflocalizedstates. . . . . . . . . . . 96
5.4.4 Blochoscillations. . . . . . . . . . . . . . . . . . . . . . . . . . 98
A Thesymplectic group Sp(2d) 103
B Themetaplecticrepresentation 105
C Notation CChapter 1
Introduction
This thesis is dedicated to the study of periodic, or close to periodic structures
in mechanics. We consider the interplay of periodic structures in classical me-
chanics and their implications on quantum mechanics. More stringently, we
consider the parallel between the properties ofquantum systemsand their clas-
sical analogs. The bridge between classical and quantum is implemented bythe
Weyl quantization, a semiclassical approach to quantum mechanics. The main
aim is the study of quantum dynamics and the spectral properties of quantum
operators.
In quantum mechanics, the time evolution is given by a unitary group that
is characterized by the (real) eigenvalues of the Hamilton operator. In essence,
one knows all about the evolution of any initial state if one has solved the cor-
responding spectral problem. This usually involves recognition and use of the
symmetries of the quantum system. While computing eigenvalues is numeri-
cally feasible, one often has to resort to more specific approximations because
ofnumericalerrors. Furthermore,itisseldom possibletoformulatequalitative
statementsabouttheevolvedstate,inparticular,afterlongtimes.
Classical mechanics usestheconcept ofpoint particles. Incontrasttoquan-
tum mechanics, this particle is coherent and localized for all times. Solutions
of the classical evolutions are obtained by integration of Hamilton’s equations
of motion. Concepts of regularity and integrability are known permitting a
characterization ofthesolutions. Integrable motionisatleastquasi–periodic.
Thequantumharmonicoscillatorhasaveryspecialevolutioninthatthepar-
ticles arepropagated in a dispersionless manner. For generic quantum systems,
it is known that the evolved state disperses which raises a legitimate question
about the dispersion of evolved states. One question we ask ourselves is: do
quantum states remainlocalized? Thelargest partof existing investigations has
beendeveloped intheopposite direction. Oneimportantresultisthequantum
ergodicity theorem [Šni74, CdV85, Zel87]. It states that almost all eigenfunc-
7

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