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On Complex Fermi Curves of Two-Dimensional Periodic Schrodinger¨ Operators

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On Complex Fermi Curves of Two-Dimensional Periodic Schrodinger¨ Operators ON COMPLEX FERMI CURVES OF TWO-DIMENSIONAL PERIODIC SCHRODINGER¨ OPERATORS Inauguraldissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften der Universit¨at Mannheim vorgelegt von Dipl.-Phys. Alexander Klauer aus Heidelberg Mannheim, 2011 Dekan: Professor Dr. Wolfgang Effelsberg, Universit¨at Mannheim Referent: Professor Dr. Martin Schmidt, Universit¨at Mannheim Korreferent: Professor Dr. Andreas Knauf, Universit¨at Erlangen-Nurnberg¨ Tag der mundlichen¨ Prufung:¨ 3. Juni 2011 Abstract In dimensions d ≥ 2, the complex Bloch varieties and the associated Fermi curves of periodic Schr¨odinger operators with quasi-periodic boundary conditions are defined as complex analytic varieties. The Schr¨odinger potentials are taken from the Lebesgue space Ld=2 in the case d > 2, and from the Lorentz{Fourier space F`1;1 in the case d = 2. Then, an asymptotic analysis of the Fermi curves in the case d = 2 is performed. The decomposition of a Fermi curve into a compact part, an asymptotically free part, and thin handles, is recovered as expected. Furthermore, it is shown that the set of potentials whose associated Fermi curve has finite geometric genus is a dense subset of F`1;1. Moreover, the Fourier transforms of the potentials are locally isomorphic to perturbed Fourier transforms induced by the handles. Finally, an asymptotic family of parameters describing the sizes of the handles is introduced. These parameters are good candidates for describing the space of all Fermi curves. Zusammenfassung In d ≥ 2 Dimensionen werden die komplexen Blochvariet¨aten und die zuge- h¨origen Fermikurven periodischer Schr¨odingeroperatoren mit quasiperiodischen Randbedingungen als komplex analytische Variet¨aten definiert. Die Schr¨odinger- potentiale entstammen im Fall d > 2 dem Lebesgueraum Ld=2 und im Fall d = 2 dem Lorentz-Fourier-Raum F`1;1. Danach wird im Fall d = 2 eine asymptoti- sche Analyse der Fermikurven durchgefuhrt.¨ Erwartungsgem¨aß erh¨alt man die Aufteilung einer Fermikurve in einen kompakten Teil, einen asymptotisch freien Teil und dunne¨ Henkel. Weiterhin wird gezeigt, dass die Menge der Potentia- le, deren zugeh¨orige Fermikurve endliches geometrisches Geschlecht hat, eine dichte Teilmenge von F`1;1 ist. Uberdies¨ sind die Fouriertransformierten der Potentiale lokal isomorph zu von den Henkeln induzierten, gest¨orten Fourier- transformierten. Schließlich wird eine asymptotische Familie von Parametern, die die Gr¨oße der Henkel beschreiben, eingefuhrt.¨ Diese Parameter sind gute Kandidaten, den Raum aller Fermikurven zu beschreiben. Contents 1 Introduction 1 1.1 The inverse problem . 1 1.2 What is done in this work . 3 1.3 The inverse problem in physics . 4 1.4 Acknowledgements . 5 2 Function spaces 7 2.1 Measurable functions and Lebesgue spaces . 9 2.2 Sobolev spaces and potential spaces . 17 2.3 Rearrangement-invariant function spaces . 22 2.4 Lorentz{Zygmund spaces . 29 2.5 Fourier spaces . 36 2.6 Bochner spaces and tensor products . 38 2.7 Other spaces . 44 2.7.1 Hardy spaces . 44 2.7.2 Banach manifolds . 47 2.8 Localisation of function space norms . 47 2.8.1 Localisation on the torus . 47 2.8.2 Localisation on discrete lattices . 52 2.8.3 Translation and levelling operators . 54 3 Schr¨odinger operators 57 3.1 Schr¨odinger operators on the torus . 57 3.2 The resolvent of the free Schr¨odinger operator . 59 3.3 Resolvents of general Schr¨odinger operators . 65 4 Fermi curves 75 4.1 Bloch varieties and Fermi curves . 75 4.2 The free Fermi curve at d =2.................... 78 4.3 Asymptotic freeness . 83 4.4 The constant potential Fermi curve at d = 2 . 88 4.5 Asymptotic moduli parameterisation . 91 4.5.1 A nonlinear perturbation of the Fourier transform . 91 4.5.2 Approximation with potentials of finite type . 104 4.5.3 Per-double-point approximation of the Fermi curve . 115 v vi CONTENTS 5 Summary and outlook 125 5.1 Works with different focus . 125 5.2 Compact resolvent . 126 5.3 Fermi curve asymptotics . 129 5.4 Where to go from here . 129 A Additional function space theory 133 A.1 Lorentz{Karamata spaces . 133 B Failed extension to Dirac operators 137 B.1 Dirac operators . 138 B.1.1 Dirac operators on the torus . 138 B.1.2 Resolvents of free Dirac operators . 139 B.1.3 Resolvents of general Dirac operators . 142 B.2 Fermi curves of Dirac operators . 145 B.2.1 Bloch varieties and Fermi curves . 145 B.2.2 Free and asymptotically free Dirac Fermi curves . 148 B.3 Asymptotic analysis of Dirac Fermi curves . 150 B.3.1 A nonlinear perturbation of the Fourier transform . 151 B.3.2 Approximation with finite-type Dirac Fermi curves . 157 B.3.3 Failure to estimate the perturbed Fourier transforms . 159 Bibliography 163 List of Figures 4.1 Cross section of R .......................... 81 4.2 Cross section of F (0) . 82 4.3 Two examples of handles . 85 vii viii LIST OF FIGURES Chapter 1 Introduction 1.1 The inverse problem The Schr¨odinger equation in d dimensions, d 2 Z>0, is given by (−∆ + u) = λψ: Here, u and are functions from Rd to C, and λ is a complex scalar. The function u acts on by left multiplication. The Laplace operator ∆, on the other hand, acts as the differential operator d X @2 ∆ = ; @x2 i=1 i d where x1; : : : ; xd are coordinates for R with respect to the canonical ordered ba- sis. Hence, for d ≥ 2, the Schr¨odinger equation is a partial differential equation. We call the operator −∆ + u the Schr¨odinger operator with potential u. In par- ticular, the Schr¨odinger equation is an eigenvalue equation for the Schr¨odinger operator in which a non-trivial solution is an eigenfunction belonging to the eigenvalue λ. The set of possible eigenvalues is the spectrum of the Schr¨odinger operator. We are interested in periodic Schr¨odinger operators. By this, we mean two things. Firstly, the potential u must be a periodic function. For d = 1, this condition has a very simple formalisation. Just pick some b 2 R, b 6= 0, and let u fulfil the condition u(x + b) = u(x) for all x 2 R: Of course, this condition does not in general remain the same if we pick a different b. In this sense, b becomes a parameter for our locus of discourse. For general d, periodicity means not just one period, but d independent periods. It is helpful to put slightly more sophistication into this concept than in the case d = 1 and express periodicity through a geometric lattice Γ ⊆ Rd of rank d: d u(x + γ) = u(x) for all x 2 R and all γ 2 Γ: 1 2 CHAPTER 1. INTRODUCTION Secondly, a solution of the Schr¨odinger equation must be a quasi-periodic function, that is, there is some boundary condition k 2 Cd such that 2πi(kjγ) d (x + γ) = e (x) for all x 2 R , γ 2 Γ; (1.1.1) where (·|·) is the complex extension of the canonical Euclidean bilinear form on Rd. This second condition is not entirely independent of the first. The periodicity of u and the chain rule imply d (−∆ + u(x)) (x + γ) = ((−∆ + u) )(x + γ) for all x 2 R , γ 2 Γ: In other words, the periodic Schr¨odinger operator commutes with all translation operators Tγ , γ 2 Γ, which are defined by (Tγ )(x) := (x + γ): Hence, assuming the Schr¨odinger operator and the translation operators are di- agonalisable in some suitable sense (take, for example, the Schr¨odinger operator as a closed operator on L2(Rd) with real-valued potential), they are simultane- ously so. Therefore, given some fixed solution , there is a function χ:Γ ! C such that the translation operators act on as Tγ = χ(γ) for all γ 2 Γ in this case. The Z-module structure of Γ then yields χ(γ1 + γ2) = χ(γ1)χ(γ2) for all γ1; γ2 2 Γ: Since the translation operators do not have zero eigenvalues, we arrive at (1.1.1), except that we posed this condition independently of the aforementioned pre- requisites. Although we introduced k as a boundary condition, we shall consider it part of an extended spectrum of the Schr¨odinger operator. That is, for each Schr¨odinger operator (identified by its potential u) we define the Bloch variety d B(u) := f(k; λ) 2 C × C : There is a non-trivial solution of the Schr¨odinger equation (−∆ + u) = λψ d such that (x + γ) = exp(2πi(kjγ)) (x) for all x 2 R , γ 2 Γg: We can now pose the inverse problem of periodic Schr¨odinger operators. It consists of two subproblems. • The moduli problem. Describe/parameterise the set of all Bloch vari- eties. • The isospectral problem. Given a fixed potential u0, describe/para- meterise the set of all potentials u such that B(u) = B(u0). As it stands, the inverse problem is somewhat ill-posed. For example, what kind of \varieties" should the Bloch varieties be, and what space should we choose u from such that B(u) actually is such a variety? However, for the purposes of this introduction, we shall use this approximate problem specification, and provide the necessary additional details later in the work. 1.2. WHAT IS DONE IN THIS WORK 3 1.2 What is done in this work It turns out that a complete solution of the inverse problem is a very extensive programme. In this work, we shall solve only a small part of it. For d = 1, the inverse problem has been solved thoroughly [MO75, MM75, MT76, Kri77, MO80, Kor08].
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