Bose-Einstein Condensation in Random Potentials

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Maximilian Pechmann

Bose–Einstein Condensation in Random Potentials

Dissertation

Fakultät für

Mathematik und

Informatik

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Bose–Einstein Condensation in Random Potentials

Maximilian Pechmann

Dissertation

Hagen, 2019

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Bose–Einstein Condensation in Random Potentials

Dissertation

zur Erlangung des Grades Doktor der Naturwissenschaften

(Dr. rer. nat.)

an der Fakult¨at f¨ur Mathematik und Informatik der

FernUniversit¨at in Hagen

verfasst von Herrn Maximilian Pechmann

aus M¨unchen

Hagen, 2019

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Erstgutachter: Prof. Dr. Wolfgang Spitzer Zweitgutachter: Prof. Dr. Robert Seiringer Tag der m¨undlichen Pr¨ufung: 15. Juli 2019

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Abstract

Bose–Einstein condensation can be understood as a macroscopic occupation of a single-particle state. But an alternative definition, which requires only a macroscopic occupation of an arbitrarily small energy band of single-particle states, is also known, referred to as generalized Bose–Einstein condensation, and considered to be thermodynamically more stable. Depending on the number of macroscopically occupied single-particle states in the condensate, three types are then distin- guished. If the number of macroscopically occupied single-particle states isfinite but at least one, then the condensation is said to be of type-I. A type-II condensate occurs if infinitely many single- particle states are macroscopically occupied. If a generalized Bose–Einstein condensation without any macroscopically occupied single-particle states is present, then one speaks of a type-III condensation. In noninteracting Bose gases, it is generally easier to prove the occurrence of generalized Bose–Einstein condensation than to determine its type. While a main step for the former task is to show that the critical density isfinite, quite detailed knowledge of the energy gaps of the corresponding single-particle Hamiltonian at the lower edge of the spectrum is needed for the latter one.

In this dissertation, we study the occurrence of a Bose–Einstein condensate in Bose gases that are placed in metrically transitive random potentials. We mainly consider the Poisson random potential. The thermodynamic limit and Dirichlet boundary conditions are used throughout this work. We explore noninteracting Bose gases in arbitrary dimension at positive temperatures and a one-dimensional interacting Bose gas at zero temperature.

We provide fairly general suﬃcient conditions for the random potential that imply the almost sure occurrence of generalized Bose–Einstein condensation in noninteracting Bose gases. These conditions are fulﬁlled by a Poisson random potential, depending on its single-impurity potential.

They are also met by the one-dimensional Luttinger–Sy model, which is a central system in the research area of Bose gases in random potentials, and a generalization thereof. Moreover, we state sufficient energy gap conditions such that one can conclude a type-I condensation in Poisson random potentials. We consider different probabilistic notions here, such as convergence in the rth mean,r PN, in probability, and with probability almost one. In particular, we prove that in the Luttinger–Sy model with infinitely strong random potential a type-I condensation occurs in the rth mean, rPN, if the particle density is greater than the critical one. For the Luttinger–Sy model with afinitely strong random potential, we show the same statement, but with probability almost one.

Finally, we discuss the Luttinger–Sy model with a repulsive contact interaction between the particles at zero temperature. We provide suﬃcient requirements such that a generalized Bose–

Einstein condensate almost surely occurs. We also show that the type of the condensate is altered by the interparticle interaction. In particular, we demonstrate a transition from a type-I to a type- III condensate when the strength of the contact interaction is increased. Furthermore, we prove that the interparticle interaction reduces the local particle density.

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Zusammenfassung

Eine Bose–Einstein-Kondensation kann als eine makroskopische Besetzung eines Einteilchen-Zu- standes verstanden werden. Jedoch ist eine alternative Deﬁnition, die als verallgemeinerte Bose–

Einstein-Kondensation bezeichnet wird, ebenfalls bekannt, setzt lediglich eine makroskopische Besetzung eines beliebig kleinen Energiebandes von Einteilchen-Zuständen voraus und gilt als thermodynamisch stabiler. Abhängig von der Anzahl der makroskopisch besetzten Einteilchen- Zustände im verallgemeinerten Bose–Einstein-Kondensat werden drei Typen unterschieden. Falls die Anzahl der makroskopisch besetzten Einteilchen-Zustände endlich und mindestens eins ist, so handelt es sich um eine Typ-I-Kondensation. Eine Type-II-Kondensation entsteht, wenn unendlich viele Einteilchen-Zustände makroskopisch besetzt sind. Falls ein verallgemeinertes Bose–

Einstein-Kondensat ohne makroskopisch besetzte Einteilchen-Zust¨ande entsteht, dann spricht man von einer Typ-III-Kondensation. Allgemein ist der Beweis einer verallgemeinerten Bose–Einstein- Kondensation in nichtwechselwirkenden Bosegasen einfacher als die Bestimmung dessen Typs.

Während ein Hauptschritt für die erste Aufgabe das Zeigen einer endlichen kritischen Dichte ist, sind für die letztere recht genaue Kenntnisse über die Abstände zwischen den Energieniveaus des entsprechenden Einteilchen-Hamiltonoperators am unteren Rand des Spektrums nötig.

Diese Dissertation besch¨aftigt sich mit dem Auftreten einer Bose–Einstein-Kondensation in Bosegasen bei Vorhandensein eines metrisch transitiven Zufallspotentials. Besonders ber¨uck- sichtigen wir hierbei das Poisson-Zufallspotential. Wir verwenden ausschließlich den thermody- namischen Limes und Dirichlet-Randbedingungen. Wir untersuchen nichtwechselwirkende Bose- gase in beliebiger Dimension bei positiven Temperaturen und ein eindimensionales wechselwir- kendes Bosegas bei TemperaturT “0.

Wir stellen recht allgemeine, hinreichende Bedingungen für das Zufallspotential bereit, die das fast sichere Auftreten einer verallgemeinerten Bose–Einstein-Kondensation in nichtwechselwirkenden Bosegasen implizieren. Diese Bedingungen werden von einem Poisson-Zufallspotential, abhängig von der Gestalt seines Einzelplatz-Potentials, erfüllt. Sie werden auch erfüllt von dem ein- dimensionalen Luttinger–Sy-Modell, das ein zentrales System im Forschungsbereich von Bosegasen in Zufallspotentialen ist, und von einer Verallgemeinerung von diesem. Zudem geben wir hinreichende Abstände zwischen den Energieniveaus an, um auf eine Typ-I Kondensation in Poisson- Zufallspotentialen schließen zu können. Dabei berücksichtigen wir mehrere Formen des Auftretens, die sich im wahrscheinlichkeitstheoretischen Sinne unterscheiden. Wir geben Bedingungen für eine Typ-I Kondensation imr-ten Mittel,rPN, in Wahrscheinlichkeit und mit Wahrscheinlichkeit fast eins an. Insbesondere zeigen wir, dass im Luttinger–Sy-Modell mit unendlich starkem Zufallspo- tential eine Typ-I Kondensation imr-ten Mittel,rPN, entsteht, falls die Teilchendichte größer als die kritische Dichte ist. Für das Luttinger–Sy-Modell mit endlich starkem Zufallspotential zeigen wir, dass dieses mit einer Wahrscheinlichkeit fast eins geschieht.

Schließlich behandeln wir das Luttinger–Sy-Modell mit einer abstoßenden Kontaktwechsel- wirkung zwischen den Teilchen bei TemperaturT “0. Wir geben hinreichende Bedingungen an, so dass eine verallgemeinerte Bose–Einstein-Konden”-sation fast sicher auftritt. Wir zeigen auch, dass der Typ des Kondensats durch die Teilchenwechselwirkung verändert wird. Insbesondere demons- trieren wir einen Übergang von einer Typ-I- zu einer Typ-III-Kondensation bei zunehmender Stärke der Kontaktwechselwirkung. Zudem beweisen wir, dass die Teilchenwechselwirkung die

¨

ortliche Teilchendichte reduziert.

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Acknowledgments

I greatly thank the FernUniversit¨at in Hagen forﬁnancial support. I am particularly grateful for the dissertation completion scholarship, which allowed me to entirely focus on my dissertation during the last six months.

I would like to express my deep appreciation and gratitude to my advisor Prof. Dr. Wolfgang Spitzer for giving me the opportunity to pursue my doctoral studies and for his constant support. I am also very thankful to Dr. Joachim Kerner for his advices and collaboration.

I am grateful to Prof. Dr. Delio Mugnolo for giving me the opportunity to work as a teaching assistance and for being such a kind and ﬂexible supervisor.

It is my pleasure to thank Prof. Dr. Werner Kirsch and Prof. Dr. Hajo Leschke for interesting discussions and useful remarks that led to an improvement of this dissertation. Last but not least, I thank my family for their support over the past years.

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1 Introduction

A Bose–Einstein condensate is a matter of state of aBose gas, that is, a particle system consisting of bosons. Its existence cannot be explained by classical means and thus is considered to be a strong indication for the validity of quantum theory. Bose–Einstein condensation occurs when a dilute Bose gas is cooled to a temperature very close to absolute zero. Particles that are in the condensate seem to have no individual properties.

Mathematically, Bose–Einstein condensation is commonly associated with a macroscopic occupation of a single-particle state, meaning that the occupation number is of the same order as the particle number N of the system in the limit N Ñ 8. In noninteracting Bose gases, that is if the bosons do not interact with each other, the condensation thermodynamically appears as a ﬁrst-order phase transition. This phase transition is unique in that it is purely a consequence of quantum statistics and not caused by interparticle interactions [Smi17].

The possibility of such a condensation was firstly discussed in 1924 by the Indian physicist Satyendranath Bose in noninteracting photons with a nonfixed particle number [Bos24]. Albert Einstein extended this idea a short time later to ideal Bose gases, that is to noninteracting Bose gases without an external potential, with afixed number of particles [Ein24, Ein25]. They realized that if one considered the particles of an ideal Bose gas to be indistinguishable, then under certain circumstances at very low but still positive temperatures they would be distributed among the energy levels of the gas such that a large portion of the bosons are in the single-particle state with the lowest energy. These particles could then no longer contribute to the pressure or the entropy of the gas. Moreover, the condensate phase would have an infinite compressibility.

These ideas initially seemed to be a mathematical curiosity with no practical beneﬁt [LSSY05].

It was only in 1938 that London proposed an explanation of the superﬂuidity experimentally discovered shortly before by means of a Bose–Einstein condensate [Lon38].

In the 1940s and 1950s, interacting Bose gases were theoretically studied at zero temperature by Bogolubov [Bog47b, Bog47a], by Penrose and Onsager [PO56], and by Beliaev [Bel58a]

for the first time. Bogolubov’s theory can accurately describe many properties of an interacting Bose gas and was extended in the 1950s and 1960 by Lee, Yang, Huang, and Luttinger [LY57, LHY57, HYL57, LY58, LY60] and others [BS57a, BS57b, Bel58b, Wu59, HP59, GA59], such as to include positive temperatures [Smi17], [LSSY05]. These approaches were, however, not rigorous and based on unproven assumptions regarding the ground state [LSSY05]. Dyson at- tempted in 1957 a mathematically rigorous proof of the asymptotic behavior of the ground-state energy of an interacting Bose gas [Dys57]. He considered the special case in which the interaction between the bosons is modeled by hard cores. The proof did not completely succeed to show the leading asymptotics of the ground-state energy, but his ideas, such as transforming hard sphere potentials into soft potentials, still have a large influence in the researchfield of interacting Bose gases [LSSY05].

In the 1960s, 1970s, and 1980s, large progress in the rigorous description of ideal Bose gases at positive temperatures has been made by van den Berg, Casimir, Girardeau, and others [Gir60, Cas68, LW79, BL82, Ber83, BLP86, BLL86]. In contrast to the common understanding

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2 1. Introduction

of Bose–Einstein condensation as a macroscopic occupation of a single-particle state, a broader deﬁnition known as generalized Bose–Einstein condensation, which requires only a macroscopic occupation of an arbitrarily narrow energy interval of single-particle states, has been introduced.

Generalized Bose–Einstein condensation is then classified into different types depending on the number of macroscopically occupied single-particle states in this condensate: Type-I or type-II condensate is present whenever finitely or infinitely many single-particle states, respectively, are macroscopically occupied. A generalized Bose–Einstein condensation without any single-particle state being macroscopically occupied is defined as type-III. It has been proved that in the simple case of an ideal Bose gas in a rectangular box of dimensions d ě 3, generalized Bose–Einstein condensation occurs if the particle density is large enough. The type, however, depends on the shape of the box or, more generally speaking, on the energy gaps of the gas at the bottom of the spectrum. In particular, a type-III Bose–Einstein condensation is possible. Furthermore, it has been argued that Bose–Einstein condensation in the generalized sense is thermodynamically more stable than in the sense of a macroscopically occupied single-particle state [Gir60, JPZ10].

The big breakthrough in the research area of Bose–Einstein condensation came in 1995, when the previously developed magneto-optical traps as well as laser and evaporative cooling techniques allowed to achieve sufficiently low temperatures, that is the nanokelvin regime, such that a Bose–Einstein condensate could be observed in experiments [AEM^`95, DMA^`95]. For these accomplishments, Cornell, Ketterle, and Wieman were awarded the 2001 Nobel Prize in Physics [CW02, Ket02]. Today, Bose–Einstein condensation is one of the most active fields of research in condensed matter physics. Interesting features of Bose–Einstein condensates include quantized vortexes and interference properties [Don91, ATM^`97]. In addition, a connection to superconductivity and superfluidity is generally accepted.

Although Bose–Einstein condensation is considered to be well-understood from a physical- theoretical point of view, its mathematical-rigorous description is still incomplete, see, for example, [LSSY05, Mic07a]. Also motivated by the successful realizations of Bose–Einstein condensations in experiments, a rigorous proof of the occurrence of Bose–Einstein condensation in interacting many- body quantum systems has been a main objective in mathematical physics over recent decades, and signiﬁcant progress has been made [LSSY05]. For instance, Dyson’s proof was completed and extended by Lieb, Seiringer, and Yngvason in the years 1998 to 2002 [LY98, LSY00, LS02]. The mathematical study of interacting Bose gases, however, remains a challenge and is mostly limited to zero temperature.

The exploration of Bose–Einstein condensation in random potentials is of particular interest [HM92, KT02, LPZ04]. It helps, for example, understanding superconductivity in porous materials [CBM^`88, HM92]. Such potentials can trigger the occurrence of generalized Bose–Einstein condensation infree Bose gases (ideal Bose gases or noninteracting Bose gases in an external potentials), see Chapter 3 and [LPZ04, KPS19b]. Knowledge about the condensate type in these cases, however, is lacking, with the exception of a few one-dimensional random models [LZ07, Jae10, KPS19b], see also Chapter 5.

One of these exceptions is a model that wasfirstly studied by Luttinger and Sy in the beginning of the 1970s [Sy72, LS73b, LS73a]. This Luttinger–Sy model is a one-dimensional, continuous, noninteracting bosonic particle system with a singular Poisson random potential of infinite strength. This model is of large interest because it is considered to be a good approximation of one-dimensional Bose gases in more general Poisson random potentials [Zag07]. Luttinger and Sy have proved that the critical density in their model is finite. Moreover, they argued that a type-I

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3

condensate occurs. Their approach, however, was mathematically not rigorous [JPZ10]. A proof for the occurrence of this type of condensation was then provided in [LZ07, Zag07].

Another interesting system is a three-dimensional, continuous random model discussed by Kac and Luttinger around the same time [KL73, KL74]. They considered a three-dimensional, free Bose gas and a repulsive interaction of ﬁnite range between the particles and randomly and uniformly distributed impurities. Kac and Luttinger have shown that, under certain conditions, generalized Bose–Einstein condensation occurs. They also presumed that in this case, solely the ground state is macroscopically occupied. To our knowledge, this Kac–Luttinger conjecture has not been proved so far, see, for example, [JPZ09, p. 6] or [LZ07, Zag07].

Although noninteracting Bose gases are a ﬁrst step to understanding Bose–Einstein condensation, one assumes that a Bose–Einstein condensate in random models generally causes a local particle density that converges to inﬁnity in the thermodynamic limit [LPZ04]. Due to this non- sensical result from a physical point of view, it is supposed that in random potentials, a repulsive interaction between the particles must not be neglected. In the research area of interacting Bose gases in random potentials, however, only a very limited amount of rigorous results is available so far. They include, for instance, [SYZ12, SW16, KPS19a].

In this work, we study continuous, bosonic particle systems in metrically transitive random potentials. We are concerned with free Bose gases in random potentials at positive temperatures in Chapters 3 to 5, and with a model of interacting bosons in a random potential at zero temperature in Chapter 6. We consider arbitrary spatial dimensions dP N in Chapters 3 and 4. The systems we discuss in Chapter 5 and 6 are one-dimensional random models.

In Chapter 2, we summarize facts and introduce concepts that are used in this thesis. We begin with measure and probability theory. In Section 2.3, we introduce random potentials. We give a short overview of the mathematical description of quantum mechanics and quantum statistics in the following two sections. In this work, we study free Bose gases solely in the grand canonical ensemble at positive temperatures, and we introduce this concept in Section 2.6. Next, we begin considering free Bose gases in random potentials. As afirst step, we explore properties of random Schrödinger operators in Section 2.7. We conclude this chapter by introducing the definitions of Bose–Einstein condensation in various probabilistic notions.

We are concerned with the occurrence of generalized Bose–Einstein condensation in free Bose gases in Chapter 3. Firstly, we shortly summarize the facts regarding the occurrence of Bose–Einstein condensation in ideal Bose gases in Section 3.1. In Section 3.2, we study the conditions that are suﬃcient to ensure the almost sure occurrence of generalized Bose–Einstein condensation in free Bose gases that are placed in random potentials.

In Chapter 4, we explore Bose gases in Poisson random potentials. These potentials are considered to be a prime example of a random potential and are commonly used to model systems with structural disorder [Mül05]. We introduce them in Section 4.1. Subsequently, we show that the requirements for the occurrence of generalized Bose–Einstein condensation from Section 3.2 are fulfilled. In Section 4.3, we investigate what kind of energy gaps are sufficient in order to prove a type-I Bose–Einstein condensation. We consider various probabilistic notations here. Lastly, we check to what extent a particular kind of Poisson random potential that can be considered very realistic from a physical point of view meets these conditions.

We study the Luttinger–Sy model without interaction between the particles at positive temperatures in Chapter 5. For this, weﬁrstly state statistical properties of Poisson random potentials on R. Subsequently, we explore in Sections 5.2 and 5.3 under what circumstances and in which

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4 1. Introduction

probabilistic notion Bose–Einstein condensation in the Luttinger–Sy model and in a generalization thereof, that is, in the Luttinger–Sy model withﬁnite interaction strength of the random potential, occurs.

In Chapter 6, we lastly introduce a contact interaction between the particles in the Luttinger–

Sy model. After defining this system in Section 6.1, we investigate in Section 6.2 under what conditions generalized Bose–Einstein condensation occurs. Moreover, we are interested in whether the interparticle interaction can reduce the local particle density if compared to the noninteracting Luttinger–Sy model. We also explore whether the strength of the interparticle interaction has an influence on the type of the condensate. We compare our results to the ones of [SYZ12], in which a similar model is studied, in Section 6.3. In Section 6.4, we finally provide miscellaneous results that we need for our proofs in Chapter 6.

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2 Preliminaries

In this chapter, we discuss facts that are helpful for understanding this dissertation. We deal with measure theory and the Stieltjes integral in Section 2.1. The fact that we can use integration by parts for Stieltjes integrals will play an important role in the proofs of Chapters 3 and 4. Next, we summarize some facts of probability theory in Section 2.2. In Section 2.3, we introduce random potentials. In the two subsequent sections, we give a short overview of the quantum mechanical description of a physical system and of quantum statistics. We are concerned with the description of a free Bose gas in the grand canonical ensemble in Section 2.6. We use this ensemble throughout this work to describe a free Bose gas at positive temperatures. We introduce random Schr¨odinger operators and state their most important general facts with respect to the occurrence of Bose–

Einstein condensation in Section 2.7. Lastly, we give the deﬁnitions of Bose–Einstein condensation in free Bose gases in Section 2.8. Parts of this chapter have been presented in my Master thesis [Pec16].

2.1 Measure Theory and Stieltjes Integral

When we deﬁne Poisson random potentials in Chapter 4, we use the notion of atoms of a measure.

We therefore begin this section with introducing their deﬁnitions by following [RS80, Section I.4]

and [Bau92]. Throughout this work, B^d is the Borel σ-algebra on R^d, d PN, and it is B “B¹. Moreover, in order to simplify the notation, we write dx “ λpdxq and also |A| :“ λpAq for a measurable set AP B^d, d PN, in the case of the Lebesgue measureλ. In addition, we write|A| for the number of elements in a ﬁnite setAĂN.

A measureM onB^dis a mapM :B^dÑ r0,8rwith the following three properties: For every A P B^d, we have MpAq ě 0. It is MpHq “0. Lastly, we have MpŤ₈

i“1Aiq “ ř₈

i“1MpAiq for every countable collection pAiq⁸i“1 withAiPB^d for everyiPN andAiXAj “ Hfor everyi‰j.

A Borel measure on B^d is defined as a measure on B^d for which MpCq ă 8 for any compact set C Ă R^d holds. We call a measure M inner regular if MpAq “ suptMpCq : C Ă A and C is compactu, outer regular if MpAq “ inftMpOq : A Ă O and O is openu, and regular if M is inner regular and outer regular. Furthermore, a measure M on B^d is called locally finite if for everyxP R^d there exists an open neighborhood O of xwith MpOqă8. A Radon measure is a measure M on B^d that is inner regular and locally finite. Lastly, a measureM onB^d is called finite if MpR^dq ă 8, and is called σ-finite if there is a countable collection of sets pAiq⁸_i“1 such that Ai PB^d andMpAiqă8 for everyi andŤ₈

i“1Ai “R^d.

Let M be a measure onB^d. Then M isabsolutely continuous with respect to the Lebesgue measureλ if and only if λpAq “0 implies MpAq “0 for any setAP B^d. Moreover,M is called singular relative to the Lebesgue measure λ if and only if there is a set A P B^d with λpAq “ 0 andMpR^dzAq “0. An atom ofM is an elementxPR^d for whichMptxuqą0.

Next, we state important general properties of measures on B^d [Bau92, §17]. Any σ-ﬁnite

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6 2. Preliminaries

measure can be uniquely written as the sum

M “Mac`Msing

whereM_ac is an absolutely continuous measure with respect to the Lebesgue measure andM_sing is a singular measure relative to the Lebesgue measure. The Radon–Nikodym theorem states that a measure M on B^d is absolutely continuous with respect to λ if and only if there exists a nonnegative, measurable function f:R^dÑRY t8usuch that

MpAq “ ż

A

fpxqdx

for every measurable setA. In this case, the functionf is uniquely deﬁned almost everywhere and is called the density ofM with respect to the Lebesgue measure.

Consequently, one possibility to construct a measure onB is the following [Bau92,§17]. Let f :RÑRY t8u be a measurable function withf ě0. Then

MpAq:“ ż

A

fpxqdx for every APB

deﬁnes a measure on B that we call the measure with density f. This measure is absolutely continuous with respect to the Lebesgue measure λ. In particular, M does not have any atoms.

In addition, we have ż

R

gpxqMpdxq “ ż

R

gpxqfpxqdx

for every measurable functiong for which either of the two integrals exists.

We discuss another possibility for defining a measure onB [Bau92,§6]. A functionf :RÑR is said to be monotonically increasing if for everyx, yPRwithxăywe havefpxqďfpyq. Suppose F : R ÑR is a monotonically increasing, left-continuous function. Then F is called ameasure- defining function. Note that a measure-defining function can have at most countable many points of discontinuity. By setting

MFpra, brq:“Fpbq ´Fpaq

for every a, bPR, aďb, a Borel measureM_F on B is uniquely defined. Moreover, the following facts are known. LetGbe another measure-defining function; then we haveM_F “M_Gif and only if G“F`cwith a constantcPR. For every Borel measureM onBthere exists a measure-defining function F such that MF “M, and we define the corresponding measure-defining function F_M by

F_MpEq:“

#Mpr0, Erq if Eě0

´MprE,0rq if Eă0 . (2.1.1)

We use the Dirac delta function δ in Chapter 4 and 5. This function can be deﬁned by a measure [Bau92, §3 Example 5]. Let c P R. Then we deﬁne the measure δ_c to be δ_cpAq “ 0 for

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2.1 Measure Theory and Stieltjes Integral 7

everyAPB withcRAandδ_cpAq “1 for every APB withcPA. We have ż

ra,bs

fpxqδ_cpdxq “

#fpcq ifcP ra, bs 0 ifcR ra, bs

if f is a measurable function f : R Ñ RY t´8,8u with |fpcq| ă 8 [Bau92, §12 Example 1].

This measure is a regular Borel measure onB [Bau92,§25]. Since δc is not absolutely continuous with respect to the Lebesgue measure, it does not posses a density with respect to the Lebesgue measure. Although an abuse of notation, it is common in physics to still use the Dirac delta functionδpx´cqas the “density” of the measure δc [Bau92,§17] and consequently obtain

ż

ra,bs

fpxqδpx´cqdx“ ż

ra,bs

fpxqδcpdxq .

We adopt this notation in this work.

There are diﬀerent notions of convergence of a sequence of measures. We state two of them in the following [Bau92,§30].

Deﬁnition 2.1.1. Let M, M1, M2, . . . be Radon measures on B. The sequence pMnqnPN

converges in the vague sense to M if and only if

nlimÑ8

ż

R

fpxqMnpdxq “ ż

R

fpxqMpdxq

for all f P C_cpRq, that is, for all continuous, compactly supported, real-valued functions on R. Now, let M, M₁, M₂, . . . be ﬁnite Radon measures on B. In this case, we say that pM_nqnPN

converges in the weak sense to M if and only if

nÑ8lim ż

R

fpxqMnpdxq “ż

R

fpxqMpdxq

for all fPC_bpRq, that is, for all bounded, continuous, real-valued functions onR.

To conclude this section, we discuss theRiemann–Stieltjes integral [HS65]. Leta, bPRwith a ăb be given. Let F, G: RÑR be measure-deﬁning functions. In addition, suppose that Gis continuous onra, bs and continuously diﬀerentiable onsa, br. Then the Riemann–Stieltjes integral ş

ra,bsGpxqMFpdxqexists [HS65, §8]. Moreover, we have [RS80, Section I.4]

ż

ra,bs

FpxqMGpdxq “ żb a

FpxqdG

dxpxqdx . (2.1.2)

Lastly, integration by parts holds, ż

ra,bs

GpxqMFpdxq “GpbqFpb`q ´GpaqFpaq ´ ż

ra,bs

FpxqMGpdxq , (2.1.3)

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8 2. Preliminaries

whereFpb`q:“lim_�Ñ0Fpb` |�|qis the right-hand limit ofF at b[HS65, Theorem 21.67].

2.2 Probability Theory

LetpΩ,A,Pqbe a probability space, that is, in particular,Pis aﬁnite measure withPpΩq “1. A real-valued random variable X is anA-B-measurable mapX :ΩÑR. X is said to beintegrable if

Er|X|s “ ż

Ω

|Xpωq|Ppdωqă8.

There are several notions of convergences for sequences of random variables [Bau02]. Suppose X, X1, X2, . . . are real-valued random variables on pΩ,A,Pq. The sequence pXnqnPN is said to P-almost surely converge to X, if

P´

nÑ8lim X_n “X¯

“1. If we have

nlimÑ8Er|X_n´X|^rs “0

where r P N, then we say that pXnqnPN converges to X in the rth mean. Lastly, the sequence pX_nqnPN converges to X in probability, if and only if

nÑ8lim Pp|Xn´X|ăηq “1 for everyηą0.

The implications of the diﬀerent notions of convergence are as follows [Bau02]. For every rąs, convergence ofpXnqnPN to X in therth mean implies convergence ofpXnqnPN toX in the sth mean. IfpXnqnPN convergences toX either in therth mean for anrPN or P-almost surely, then pX_nqnPN also converges to X in probability. Finally, theVitali convergence theorem implies the following. If pX_nqnPN converges to X in probability and there exist an r P N and a random variable Y such thatEr|Y|^rsă8 andPp|Xn|ďYq “1 for every nPN, thenpXnqnPN converges to X also in therth mean [Bau92,§21].

An important tool in probability theory is the Borel–Cantelli lemma. We state it in the next theorem [Bau02, Lemma 11.1].

Theorem 2.2.1. (Borel–Cantelli lemma)

LetpΩ,A,P) be a probability space, and pA_nqnPN be a sequence of measurable sets. We deﬁne the set

A₈:“lim sup

nÑ8 A_n “ č⁸

n“1

ď8 i“n

A_i“ tωPΩ:ωPA_n for inﬁnitely many nu . If ř₈

n“1PpAnq ă 8, then one concludes PpA₈q “ 0. In addition, if pAnqnPN is a collection of pairwise independent sets, the condition ř₈

n“1PpA_nq “ 8 implies PpA₈q “1.

Lastly, we mention the strong law of large numbers. In Section 2.3, we will give a similar statement to Theorem 2.2.2 that is valid under more general circumstances, see Theorem 2.3.3. Let

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2.3 Random Potentials 9

pX_nqnPN be a sequence of integrable, real-valued random variables, andErX_nsbe the expectation value ofXn for eachnPN. Then we say that pXnqnPN satisﬁes thestrong law of large numbers if

nlimÑ8

1 n

ÿn i“1

`Xi´ErXis˘

“0

P-almost surely [Bau02, Deﬁnition 10.1]. For example, this law applies to a sequencepXnqnPN of pairwise independent, identically distributed, real-valued random variables, as the next theorem shows [Bau02, Theorem 12.1].

Theorem 2.2.2. Let pX_nqnPN be a sequence of integrable, identically distributed, pairwise independent, real-valued random variables. Then pX_nqnPN satisﬁes the strong law of large numbers, that is, we P-almost surely have

nlimÑ8

1 n

ÿn i“1

X_i“ErX₁s .

2.3 Random Potentials

In this work, we consider random potentials that are metrically transitive. They are used to describe disordered matter like amorphous solids [LMW03]. A special property of such a random potentialV is that the corresponding random variablesVpxq, Vpyqat two different pointsx, yPR^d, x‰y, are, although possibly not stochastically independent, almost stochastically independent in the sense that the correlation between Vpxq andVpyqvanishes with increasing distance between both pointsx, y. In order to define metrically transitive random potentials in Definition 2.3.2, we firstly introduce the notions of random fields and metrically transitive groups of automorphism on a probability spacepΩ,A,Pq. We follow [FP92, Chapter 1].

LetGbe a set. A functionV :ΩˆGÑRis said to be a random function ifVp¨, gq:ΩÑR is a real-valued random variable for every gP G. V is called a random ﬁeld, if Gis either R^d or Z^d, d P N, and is, together with a suitable operation, an Abelian group. A random ﬁeld is also called arandom process orstochastic process ifd“1. Note that we meanVp¨, gq if we writeVpgq in the rest of this work.

We call a measurable automorphism T on pΩ,A,Pqmeasure-preserving ifPpT^´¹Aq “PpAq for every A P A. A topological group T of measurable automorphisms on pΩ,A,Pq is called stochastically continuous if lim_T₁_Ñ_T PpT₁A�T Aq “ 0 for every A P A, T P T. Here, X�Y denotes the setpXYYqzpX XYq for X, Y P A. Finally, a stochastically continuous group T of measure-preserving automorphisms onpΩ,A,Pq is calledmetrically transitive if for every T PT and everyAPA we have the following implication: FromT A“A followsPpAq P t0,1u. We now state the deﬁnitions of random potentials and of metrically transitive randomﬁelds, and hence of metrically transitive random potentials.

Deﬁnition 2.3.1. Let V : Ω ˆR^d Ñ R be a random ﬁeld. Moreover, we assume that the realizations Vpω,¨q : R^d Ñ R are P-almost surely locally square-integrable functions, or that d“1 and the realizations areP-almost surely of the formVpω,¨q “ř

jPZσδp¨´xˆjpωqqwithσą0 or σ “ 8, δ the Dirac delta function, ˆxjpωq P R for every j P Z, and ˆxjpωq ‰ xˆipωq for every i, j PZ withi‰j. ThenV is called a random potential.

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10 2. Preliminaries

Deﬁnition 2.3.2. Let V : ΩˆR^d ÑR be a random ﬁeld. We call V metrically transitive if there exists a stochastically continuous, metrically transitive group tTg : g P Gu of measurable automorphisms on pΩ,A,Pqsuch that we haveVpω, g`g₁q “VpT_g₁ω, gqfor everyg, g₁ PGand everyωPΩ.

In the rest of this work, we mean metrically transitive random potentials if we speak of random potentials. In addition, if we require a property such as Er|Vp0q|^js ă 8 for a j P N, we always mean a random potential whose realizations are P-almost surely functions. A central property of metrically transitive randomﬁelds is that a statement similar to Theorem 2.2.2 is still true [FP92, Proposition 1.13].

Theorem 2.3.3. LetV be a metrically transitive randomﬁeld such that Er|Vp0q|s “ż

Ω

|Vpω,0q|Ppdωqă8 .

Moreover, let Λ Ă R^d be a cube of volume |Λ| that is centered at zero. Then we P-almost surely have

|Λlim|Ñ8

1

|Λ| ż

Λ

Vpω, gqdg“ErVp0qs .

In particular, if G“Z, then one P-almost surely has

nlimÑ8

1 2n`1

ÿn g“´n

Vpω, gq “ErVp0qs .

The most important examples of metrically transitive random potentials from a physical point of view include thePoisson random potential. Bose–Einstein condensation in Bose gases that are placed in a Poisson random potential can occur under certain conditions. We will examine these facts in detail in Chapter 3 and Section 4.2, see also the beginning of Chapter 4.

2.4 Quantum Mechanics

We give a short overview of the main facts of quantum mechanics in this section and of quantum statistics in the next section that are required or useful to understand this work. We start by formulating the most important postulates of quantum mechanics [Neu96]. A (pure) state of an isolated quantum system is described by a vector of a complex separable Hilbert space, which is also called thestate space. Everyobservable, that is a measurable physical quantity, of a system is represented by a self-adjoint, linear operator acting on the considered state space. Possible values of the observable are elements of the spectrum of the corresponding operator.

The Sobolev space H¹pΛq and its subspace H₀¹pΛq are two examples of Hilbert spaces that are particularly important in Quantum mechanics. We start by deﬁning the Hilbert spaceH¹pΛq and follow [LL01]. LetΛĂR^d be an open set. The set

H¹pΛq:“!

f :ΛÑC:f PL²pΛqand∇f PL²pΛq)

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2.4 Quantum Mechanics 11

equipped with the inner product xf, gyH¹pΛq:“ż

Λ

fpxqgpxqdx` ÿd n“1

ż

Λ

Bfpxq Bx_n

Bgpxq Bx_n dx

for everyf, g PH¹pΛq is a Hilbert space. The norm of an element inH¹pΛq consequently reads

}f}H¹pΛq“

¨

˝ż

Λ

|fpxq|² dx` ż

Λ

|∇fpxq|² dx

˛

‚

1{2

.

We haveH¹pΛqĂL²pΛq for every open setΛĂR^d.

Note that the differentiations shall be understood in the distributional sense. In particular, the meaning of∇f PL²pΛqis as follows. We introduce thespace of test functionsDpΛq. This space is defined as the set C_c⁸pΛq, that is the set of all infinitely differentiable functions with compact support in Λ, together with the following notion of convergence. A sequencepϕ_mqmPN ĂC_c⁸pΛq is said to convergence inDpΛqto a functionϕPC_c⁸pΛqif and only if the following two conditions hold: Firstly, there exists a compact setC Ă Λsuch that the support of ϕ_m´ϕis a subset of C for everymPN. Secondly, for any nonnegative integer numbersα1, . . . ,α_dPNY t0uwe have

mlimÑ8 sup

x“px1,...,xdqPC

ˇˇ ˇˇ

ˆ B Bx₁

˙α1

. . . ˆ B

Bx_d

˙αd

ϕ_mpxq ´ ˆ B

Bx₁

˙α1

. . . ˆ B

Bx_d

˙αd

ϕpxq ˇˇ ˇˇ“0.

Now, ∇f represents a collection ofd functionsb₁, . . . , b_d PL²pΛq for which we have ż

Λ

fpxqBϕ

Bxnpxqdx“ ´ż

Λ

b_npxqϕpxqdx

for all n“1, . . . , dand for all ϕPDpΛq.

The Hilbert spaceH¹pΛqcan also be deﬁned as the completion ofC⁸pΛq with respect to the H¹pΛq-norm. Furthermore, the setC_c⁸pR^dqis dense inH¹pR^dqwith respect to theH¹pR^dq-norm.

For Λ Ĺ R^d, the Hilbert space H₀¹pΛq is deﬁned as the completion of C_c⁸pΛq with respect to the H¹pΛq-Norm. The space H₀¹pΛq is a subspace of H¹pΛq. It turns out that H₀¹pΛq is suitable for the discussion of diﬀerential equations with Dirichlet boundary conditions, which is a main concern in this work. We also note that ifI “ sa, bris a bounded open interval, the setH¹pIqand, consequently, the set H₀¹pIqconsist only of functions that are continuous onra, bs.

To construct a self-adjoint operatorAon a Hilbert spaceH, it is often convenient toﬁrstly deﬁne a symmetric operator onC_c⁸pΛqand subsequently to extend it to a self-adjoint operator. In fact, if an operator Aon a Hilbert space H is essentially self-adjoint, then there exists a unique, self-adjoint extension ofA onH [Wei00, Chapter 2.4].

Although a symmetric operator A is always closable with A^˚˚ being the closure of A, in general none of the closed extension ofAare necessarily self-adjoint [RS80, p. 279]. If an operator A is semi-bounded and symmetric, the existence of at least one self-adjoint extension of A is guarantied [Wei00, Theorem 4.15]. One might face, however, the problem that several self-adjoint extensions exist. A possibility to characterize a unique self-adjoint extension is through quadratic forms [RS80, Section VIII.6], [Wei00, Section 4.2]. We state the details in the next two theorems.

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12 2. Preliminaries

For the convenience of the reader, we brieﬂy recall the deﬁnitions of quadratic forms and their most important properties.

Let H be a Hilbert space. A map q : Qpqq ˆQpqq Ñ C with Qpqq a dense linear subset of H such that qp¨,ψq is conjugate linear and qpϕ,¨q is linear for every ϕ,ψ P Qpqq is called a quadratic form on H. In this case, Qpqq is said to be form domain. We call q symmetric if qpϕ,ψq “qpψ,ϕqfor everyϕ,ψPQpqq. A quadratic formqis said to benonnegative ifqpϕ,ϕqě0 for every ϕ P Qpqq. If there exists an M P R such that qpϕ,ϕq ě ´M}ϕ}²_H for every ϕ P Qpqq then we call q semi-bounded and M the lower bound of q. Moreover, a semi-bounded quadratic form q with lower bound M is said to be closed if its domain Qpqq is complete under the norm }ψ}`1:“b

qpψ,ψq ` pM `1q}ϕ}²_H.

In addition, we call a linear operator A : H Ñ H with domain DpAq symmetric, if A is densely deﬁned and we have xAϕ,ψyH “ xϕ, AψyH for every ϕ,ψ P DpAq. We call A semi- bounded if A is symmetric and if there exists an a P R such that xϕ, AϕyH ě axϕ,ϕyH for every ϕ P DpAq. In particular, the operator A is said to be nonnegative if A is symmetric and xϕ, AϕyH ě0 for every ϕPDpAq.

Theorem 2.4.1. If q is a closed, semi-bounded, quadratic form on a Hilbert space H, then q is the quadratic form of a unique, semi-bounded self-adjoint operator on H.

Theorem 2.4.2. (Friedrichs)

Let A be a nonnegative operator on a Hilbert space H with domain DpAq. We deﬁne qpϕ,ψq:“ xϕ, Aψy for every ϕ,ψ P DpAq. Thenq is a closable, nonnegative quadratic form on H, and the closure q¯of q is the quadratic form of the unique Friedrichs extension AF of A. AF is unique in that it is the only self-adjoint extension ofA whose domain is a subset of the domain ofq.¯ A_F is nonnegative as well.

One of the most important operators in quantum mechanics in general and for this work in particular is theDirichlet–Laplacian ´�^DΛ on a boundedregion, that is, a bounded, measurable, open, connected, and nonempty setΛĹR^d [RS78, GT01]. This operator is deﬁned onL²pΛqwith domain H₀¹pΛq. It is the unique self-adjoint extension of the nonnegative, symmetric operator

´�r^DΛ :“ ´ ÿd n“1

B² Bx²_n

onL²pΛqwith domainDp´�r^DΛq:“C_c⁸pΛq. To be more speciﬁc, the Dirichlet–Laplacian is deﬁned as the unique self-adjoint operator onL²pΛq whose quadratic form is the closure of the quadratic form

qpf, gq “ ż

Λ

∇fpxq ¨∇gpxqdx

with domain C_c⁸pΛq. Its normalized eigenfunctions form an orthonormal basis of L²pΛq. The spectrum of ´�^D_Λ is purely discrete and consists only of positive, isolated eigenvalues of ﬁnite multiplicities [RS78, Section XIII].

The Dirichlet–Laplacian is part of theHamiltonian. This latter operator corresponds to the observable of the total energy of a particle system and is one of the most important operators in quantum mechanics as well. If one considersN PNparticles that are conﬁned in a bounded region

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2.4 Quantum Mechanics 13

ΛĹR^d, are placed in an external potentialV : ΛÑR, and interact with each other through an interparticle interactionW :Λ^N ÑR, theN-particle Hamiltonian H^pNq_Λ reads

H^p_Λ^N^q:“ ÿN j“1

´´�^D,_Λ^p^j^q¯

` ÿN j“1

Vpx^p^j^qq `Wpx^p¹^q, x^p²^q, . . . , x^p^N^qq (2.4.1)

withx^p¹^q, x^p²^q, . . . , x^p^N^qPΛand is deﬁned on theN-particle Hilbert space H_Λ^pNq:“â^N

L²pΛq »L²pΛ^Nq. whereÂ_N

L²pΛqis theN-fold tensor product ofL²pΛq. Here,´�^D,_Λ^p^j^qis theDirichlet–Laplacian on Λ with respect to thejth particle and deﬁned as the unique self-adjoint extension of the operator ´řd

n“1B²{pBx^pn^j^qq² onL²pΛq with domainC_c⁸pΛq. The operator´�^D,Λ^p^j^q corresponds to the observable of the kinetic energy of thejth particles. In order to simplify the notation, we have set

�“1 for the Planck constant andm“1{2 for the mass of the particles throughout this work.

Note that if the particles of the system do not interact with each other and, consequently, the N-particle Hamiltonian (2.4.1) does not posses the last term, it is suﬃcient to study the corresponding single-particle Hamiltonian H^p1q_Λ . We will explain this fact in more detail in Section 2.6.

Nevertheless, the question arises how to deﬁne a self-adjoint operator that is the sum of operators. One can deﬁne the sum A`B of two self-adjoint operatorsAwith domain DpAqand B with domainDpBqon a Hilbert space as

pA`Bqf :“Af`Bf for everyf PDpA`Bq:“DpAq XDpBq .

There are, however, cases for whichDpA`Bqconsists only of the zero vector. Hence, we also state the following possibility for deﬁning the operator A`B [BB93, p. 236]. We remark that if A is nonnegative and self-adjoint, then the square rootA¹^{² ofAis uniquely deﬁned as a nonnegative, self-adjoint operator [Wei00, Theorem 8.22].

Deﬁnition 2.4.3. Let A with domain DpAq and B with domain DpBq be two nonnegative, self-adjoint operators on a Hilbert space H . Suppose that the set D0 :“ DpA¹^{²q XDpB¹^{²q is dense inH, and that the nonnegative, quadratic form

q:D0ˆD0 ÑC, pϕ,ψq ÞÑ xA¹^{²ϕ, A¹^{²ψyH ` xB¹^{²ϕ, B¹^{²ψyH

is well defined on D0. Then the closure ¯q of q is a nonnegative, densely defined quadratic form with domain Qpq¯q Ą D0. The nonnegative, self-adjoint operator A`B with domain DpA`Bq that is uniquely defined by the conditions

DpA`BqĂQpq¯q and xϕ,pA`BqψyH :“q¯pϕ,ψq for all ϕPQpq¯q,ψPDpA`Bq is called theform sum ofA andB.

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14 2. Preliminaries

2.5 Quantum Statistics

In order to simplify the quantum mechanical description of many-body systems such as Bose gases, one can characterize a system by amacroscopic state, which is deﬁned by only a few quantities such as the temperature or the pressure. In general, there are manymicroscopic states that correspond to a given macroscopic state. A density matrix is an incoherent superposition of microscopic states and represents the collection of these states. It includes all physically relevant information of the particle system [Hua87, Sak06].

Density matrices are a special type oftrace class operators. For the convenience of the reader, we begin by introducing their deﬁnition. We denote the space of all linear bounded operators on a Hilbert space H byLpHq. For the rest of this section, we assume that H is a separable Hilbert space, and thatpϕ^jqjPNis an orthonormal basis ofH. Thetraceof a nonnegative operator APLpH qis deﬁned as Tr_HrAs:“ř₈

j“1xϕ^j, Aϕ^jyH [RS80, Chapter VI]. Note that this value is independent of the chosen orthonormal basis. Moreover, if AP LpHq, then the operator |A| :“

?A^˚Aexists. Now, the space of all trace-class operators onH, tAPLpH q: Tr_Hr |A| să8u, is denoted byL¹pHq. Finally, we call a self-adjoint, nonnegative trace-class operatorD PL¹pHq a density matrix, if Tr_H rDs “1.

A convenient fact of density matrices is that they can be written in a particular simple form, by the spectral theorem: Let D P L¹pHq be a density matrix. Then there exists a sequence pλ^jqjPN Ă R and an orthonormal basis pϕ^jqjPN of H with the following properties. The set pϕ^jqjPN is an eigenbasis corresponding to the operatorD. The numbersλ^j are the eigenvalues of D. We have 0ďλ^j ď1 for everyjPN andř

jPNλ^j “1. Lastly, one has D “ ÿ

jPN

λ^j|ϕ^jyxϕ^j| ,

where|ϕ^jyxϕ^j| is the orthogonal projection onto the subspace spanned by the vector ϕ^j.

Finally, Deﬁnition 2.5.1 and Theorem 2.5.2 show that the expression Tr_HrADs whereA is a self-adjoint operator and D is a density matrix is particularly useful. Note here that if A a linear bounded operator on H, then the trace of DA is ﬁnite, Tr_HrDAs “ Tr_HrADs ă 8 [RS80]. On the other hand, if Ais a linear unbounded operator on H, D “ř

jPNλ^j|ϕ^jyxϕ^j| is a density matrix, and ř

jPNλ^jxϕ^j, Aϕ^jyH is convergent, then we set Tr_HrADs :“ Tr_HrDAs :“ ř

jPNλ^jxϕ^j, Aϕ^jyH ă8 [LS10].

Deﬁnition 2.5.1. Let H be a separable Hilbert space, A a self-adjoint operator on H , and D “ř

jPNλ^j|ϕ^jyxϕ^j| a density matrix. Then we call xAyD :“ ÿ

jPN

λ^jxϕ^j, Aϕ^jyH

theensemble average of Awith respect to a system in the stateD.

Theorem 2.5.2. Let H be a separable Hilbert space, A a self-adjoint operator on H, and D “ ř

jPNλ^j|ϕ^jyxϕ^j| a density matrix. Moreover, we assume that ř

jPNλ^jxϕ^j, Aϕ^jyH ă 8 or that Ais bounded. Then we have

xAyD “Tr_HrDAs “Tr_HrADsă8 .

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2.6 The Free Bose Gas in the Grand Canonical Ensemble 15

2.6 The Free Bose Gas in the Grand Canonical Ensemble

We now study a free Bose gas with a given particle numberNPNthat are trapped in a bounded region Λ_N Ĺ R^d, |Λ_N| ă 8. Note that we have introduced the subscription N here since we will eventually scale Λ with the particle number N. It is convenient to use the grand canonical ensemble for this task [LSSY05]. In this statistical ensemble, neither the energy nor the particle numberNr PN of the system isﬁxed. Instead, the ensemble average of the particle number is set to be equal to N. We introduce the suitable Hilbert space for this description as a ﬁrst step and follow [RS80, Chapter II.4]. To simplify the notation, we set

|jy:“ϕ^j if ϕ^j is a state (ϕ^jPH, }ϕ^j}H “1), and

|j1j2 . . . jNy:“ϕ^j¹ bϕ^j²b. . .bϕ^j^N if ϕ^j^k, k“1, . . . , N, are states.

Let p|jyqjPN be an orthonormal basis ofH_Λ^p1q_N, that is, the single-particle Hilbert space with respect toΛ_N. Thenp|j₁ . . . j_N_ryqj1,...,j_N_ĂPN is an orthonormal basis ofH_Λ^p_N^N^r^q. We call

H_Λ^p_N^N^r^q^,sym:“

Nr

â

s

L²pΛ_Nq “S_N_rH_Λ^p_N^N^r^q (2.6.1) the symmetric N-particle Hilbert space with respect tor Λ_N. Here, S_N_r :“ _N!_r¹ ř

πPPNĂ is the sym- metrizer on H_Λ^p_N^Nq^r , and ÂNr

s L²pΛ_Nq is the Nr-fold symmetric tensor product of L²pΛ_Nq. The Hilbert spaceH_Λ^p_N^N^r^q^,symis separable and used to describe a bosonic system that consists of aﬁxed particle number NrPN.

Next, the bosonic Fock space with respect toΛN is deﬁned as the set F_Λ^sym_N :“ à⁸

Nr“0

H_Λ^p_N^Nq,sym^r

together with the inner product

x ¨ | ¨ y_F_Λ^sym

N :“ ÿ⁸

Nr“0

x ¨ | ¨ y

H_Λ^p^N_N^Ă^q,sym

onF_Λ^sym_N ˆF_Λ^sym_N . Here, we have set H_Λ^p_N⁰^q^,sym :“C. The space F_Λ^sym_N is a separable Hilbert space and used to describe particle systems with a nonﬁxed number of bosons. An orthonormal basis of FΛ^symN is given by

�p1,0,0, . . .q,p0,|j11ysym,0,0, . . .q,p0,0,|j21j22ysym,0,0, . . .q, . . .(

j₁₁,j₂₁,j₂₂...PN

Bose-Einstein Condensation in Random Potentials

Maximilian Pechmann

Bose–Einstein Condensation in Random Potentials

Dissertation

Fakultät für

Mathematik und

Informatik

Bose–Einstein Condensation in Random Potentials

Bose–Einstein Condensation in Random Potentials

Abstract

Zusammenfassung

Acknowledgments

Contents

1 Introduction

2 Preliminaries

2.1 Measure Theory and Stieltjes Integral

2.2 Probability Theory

2.3 Random Potentials

2.4 Quantum Mechanics

2.5 Quantum Statistics

2.6 The Free Bose Gas in the Grand Canonical Ensemble