SPECTRAL PROPERTIES OF RANDOM BLOCK OPERATORS JACOB W. CHAPMAN GÜNTER STOLZ, COMMITTEE CHAIR LEONARD CHOUP BHARAT SONI RUDI WEIKARD ZHIJIAN WU

SPECTRAL PROPERTIES OF RANDOM BLOCK OPERATORS by JACOB W. CHAPMAN GÜNTER STOLZ, COMMITTEE CHAIR LEONARD CHOUP BHARAT SONI RUDI WEIKARD ZHIJIAN WU A DISSERTATION Submitted to the graduate faculty of The University of Alabama, The University of Alabama at Birmingham, and the University of Alabama at Huntsville, in partial fulfillment of the requirements for the degree of Doctor of Philosophy BIRMINGHAM, ALABAMA 2013

SPECTRAL PROPERTIES OF RANDOM BLOCK OPERATORS JACOB W. CHAPMAN APPLIED MATHEMATICS ABSTRACT Ever since the introduction of the Anderson model in 1958, physicists and mathematicians alike have been interested in the effects of disorder on quantum mechanical systems. For example, it is known that transport is suppressed for an electron moving about in a random environment, which follows from localization results proven for the Anderson model. Quantum spin systems provide a relatively simple starting point when one is interested in studying many-body systems. Here we investigate a random block operator arising from the anisotropic xy-spin chain model. Allowing for arbitrary nontrivial single-site distributions, we prove a zero-velocity Lieb-Robinson bound under the assumption of dynamical localization at all energies. After a preliminary study of basic properties and location of the almost-sure spectrum of this random block operator, we apply a transfer matrix formalism and prove contractivity and irreducibility properties of the Fürstenberg group and, in particular, positivity of Lyapunov exponents at all nonzero energies. Then in the general setting of random block Jacobi matrices, we establish a Thouless formula, and under contractivity and irreducibility assumptions, we conclude dynamical localization via multiscale analysis by proving a Wegner estimate and an initial length scale estimate. Finally we apply our general results to prove localization for the special case of the Ising model, and we discuss a critical energy that arises. ii

DEDICATION I dedicate this dissertation to my loving wife, Marla, and my family and friends. Their support and encouragement have meant a great deal to me and has kept my motivation strong. I also dedicate this to the late Dr. James Ward, who mentored me as an undergraduate and helped shape the beginning of my mathematical career. Most importantly, I dedicate this to my personal Savior, Jesus Christ, without whom I would be lost and certainly would not have been equipped to complete this work. iii

ACKNOWLEDGMENTS I would like to thank my committee members for their service and the giving of their time to see my education through to the end. I can never sufficiently thank my advisor, Dr. Günter Stolz, for all his support and the many hours he spent answering my questions and emails, typing up lengthy files to explain new ideas, teaching classes and seminars, and just in general giving so freely of his time and knowledge. I would also like to thank the many professors I have had who have shaped my education, the UAB Mathematics Department for the numerous opportunities I was given, and the office staff for all the support and help in making everything run smoothly. iv

TABLE OF CONTENTS Page ABSTRACT.....................................................................ii DEDICATION.................................................................. iii ACKNOWLEDGMENTS....................................................... iv LIST OF FIGURES............................................................ vii CHAPTER 1 INTRODUCTION........................................................ 1 2 QUANTUM SPIN SYSTEMS AND THE MODEL........................ 8 2.1 The xy-spin Chain Model........................................ 8 2.2 Lieb-Robinson Bound........................................... 14 2.3 Emergence of the Random Block Operator...................... 19 3 BASIC SPECTRAL PROPERTIES......................................22 4 ALMOST-SURE SPECTRUM........................................... 25 4.1 The Anderson Model............................................25 4.2 A Periodic Support Theorem.................................... 26 4.3 Inclusions for the Spectrum of H ω,γ............................. 32 5 POSITIVITY OF LYAPUNOV EXPONENTS........................... 45 5.1 Background..................................................... 45 5.1.1 Lyapunov Exponents.................................... 45 5.1.2 p-contractivity and L p -Strong Irreducibility............. 49 5.2 Transfer Matrices............................................... 50 5.2.1 Standard Transfer Matrices.............................. 51 5.2.2 Modified Transfer Matrices.............................. 52 5.2.3 The Fürstenberg Group and the Gol dsheid-margulis Criterion................................................ 53 5.3 Positivity of Lyapunov Exponents at Nonzero Energies.......... 55 v

TABLE OF CONTENTS (Continued) Page 5.4 Positivity of Lyapunov Exponents at Zero Energy............... 64 5.4.1 G(0) is not Zariski-Dense in Sp 2 (R)...................... 64 5.4.2 Reduction to the Anderson Model....................... 66 6 A THOULESS FORMULA.............................................. 70 6.1 Ergodic Block Jacobi Matrices.................................. 71 6.2 A Thouless Formula.............................................73 7 DYNAMICAL LOCALIZATION.........................................82 7.1 Main Assumption and Theorem................................. 83 7.2 Wegner Estimate................................................87 7.2.1 Transfer Matrices and Bounds on Green s Function...... 87 7.2.2 Wegner Estimate........................................ 90 7.3 An Initial Length Scale Estimate................................ 98 8 THE ISING MODEL CASE............................................ 101 LIST OF REFERENCES......................................................107 APPENDIX A FLOQUET THEORY FOR 1-D SCHRÖDINGER AND BLOCK OPERATORS......................................................... 111 A.1 1-D Schrödinger Operators.................................... 111 A.1.1 Periodic Functions..................................... 111 A.1.2 Momentum Representation of the Periodic Schrödinger Operator...............................................113 A.1.3 Restrictions of H = + V with Quasiperiodic Boundary Conditions.................................. 118 A.2 Periodic Block Schrödinger Operators......................... 122 B GREEN FUNCTION FORMULA...................................... 127 C TECHNICAL LEMMAS............................................... 138 vi

LIST OF FIGURES Figure Page 1 Geometry behind σ(h c,γ ) for c > 2 and small γ.............................. 35 2 Geometry behind σ(h c,γ ) for c > 2 and large γ...............................36 3 Geometry behind σ(h c,γ ) for 0 c 2...................................... 37 vii

1 CHAPTER 1 INTRODUCTION Motivated by the Nobel Prize-winning works of Anderson [2] and the discovery of quasicrystals by Shechtman [40] in the early 1980s, the theory of random operators and quasiperiodic operators has exploded with productivity and become an important field within mathematical physics. In 1958 Anderson introduced a model, now called the Anderson model, which describes the quantum mechanical effects due to the inherent disorder present in materials such as alloys and amorphous media. Anderson noticed that electron transport is suppressed due to disorder, a phenomenon which we call Anderson localization. This concept has kept physicists and mathematicians busy ever since its discovery. It has been proven mathematically that the one-dimensional Anderson model exhibits localization at arbitrary disorder and at all energies, whereas the multidimensional Anderson model is localized (i) at all energies only for sufficiently large disorder, and (ii) near band edges of the spectrum at arbitrary disorder. The Anderson model has been intensely studied over the past several decades, and several methods of proving localization have been developed. The first was provided in 1977 for a related one-dimensional model by Gol dsheid, Molchanov, and Pastur [24]. A few years later, Kunz and Souillard [36] proved it for the actual one-dimensional Anderson model. In 1983 Fröhlich and Spencer [18] developed a method known as multiscale analysis (MSA), which has been streamlined and improved upon by many

2 experts over the years, ultimately culminating in the process known as bootstrap MSA, which was introduced by Germinet and Klein [21] and currently yields the strongest form of localization possible through MSA. (The literature on MSA is vast, so we refer to the review [31] for a more complete history.) Two great benefits in using MSA are that it works in arbitrary dimension and that it allows for arbitrary nontrivial single-site distributions (e.g. Bernoulli-type distributions). In 1993 Aizenman and Molchanov [1] introduced the fractional moment method (FMM), a method which is remarkably less technical than MSA though it requires smoothness assumptions on the single-site distribution. The FMM also works in arbitrary dimension and gives an even stronger form of dynamical localization than MSA, i.e. true exponential decay rather than subexponential. (We refer to the review [42] for more information on the FMM.) However, there are instances (e.g. non-monotone models, models with discrete distributions) where it is difficult or impossible to make use of the FMM, and one typically turns to MSA. In recent years there has been an interest in random block operators, motivated by various applications in physics. Kirsch, Metzger, and Müller [30] study spectral properties of block operators arising from disordered systems such as dirty semiconductors. This investigation was continued in [19, 20], where bootstrap MSA is used to conclude localization for these models. Hamza, Sims, and Stolz [26] have also considered block operators which arise from the study of the xy-spin chain, which goes back to the seminal work of Lieb, Schultz, and Mattis [37]. In [26] the authors consider a random exterior magnetic field and prove localization in the isotropic case, reducing the problem in some sense to that of the Anderson model. They go on

3 to prove a Lieb-Robinson bound and decay of ground-state correlations as a result. Moreover, the authors suggest the extension of their results to the anisotropic case, which in particular gives rise to a non-monotone random block operator. This was the motivating factor behind this dissertation research. One interesting development in this direction has been in the recent work of Elgart, Shamis, and Sodin [16], where the authors consider a more general class of block operators, which includes the anisotropic xy model as a special case. In the large disorder regime, the authors prove dynamical localization via a modification of the FMM. Because the one-dimensional Anderson model appears in the xy-model, one expects localization to hold also with small (fixed) disorder and at all energies by using the same methods which are used for the 1-D Anderson model, e.g. positivity of Lyapunov exponents and multiscale analysis. One benefit, as mentioned above, is that we can assume general nontrivial single-site distributions, whereas in [16] some smoothness is needed to make the FMM work. We give in Chapter 2 a brief introduction to quantum spin systems and in particular the anisotropic xy-spin chain Hamiltonian which is introduced in [37]. We then show, as explained in [26], how such Hamiltonians may be diagonalized, using a Jordan-Wigner transform, in terms of a system of operators satisfying the canonical anticommutation relations (CAR), thus resulting in the random block operator which we shall be interested in. Such a block operator is then shown to be unitarily equivalent to a block Jacobi matrix, which will be more conducive to the transfer matrix formalism which is applied in Chapters 5 and 7. By modifying arguments from [26], we show that a (subexponential) zero-velocity Lieb-Robinson bound holds under the

4 assumption that the block operator is subexponentially dynamically localized at all energies. A preliminary investigation into the nature of a more general class of block operators is done in Chapter 3, and we discuss basic spectral properties which correspond to results in [30] which pertain to a similar class of operators. In particular, we show symmetry of the spectrum about 0, a simple bound for the spectrum, and a condition which implies the existence of a spectral gap. Such results are used in Chapter 4 while studying the almost-sure spectrum of the block operator arising from the anisotropic xy-spin chain. While an explicit formula for the spectrum is not obtained in the most general setting, a periodic support theorem is established (similar to [28]), which states essentially that the almost-sure spectrum consists of all the spectra of operators with periodic potentials taking values within the range of the random parameters. Proving such a theorem relies on some results from Floquet theory, which were adapted from [35] and placed in Appendix A. We then show just how much spectrum is generated by the constant potentials, which, in the case of the Anderson model, is all of the spectrum. In a special case, we show that this is indeed all of the spectrum for the random block operator. In Chapter 5 we first present some background taken from [4, 11] on Lyapunov exponents, symplectic matrices, exterior products p R N of R N, and the concepts of p-contractivity and L p -strong irreducibility. These objects and concepts are of higher generality than what one needs for the 1-D Anderson model because the transfer matrices we encounter are 4 4. We discuss the standard transfer matrices and how to modify them to make them symplectic, which is needed to ensure that the

5 Lyapunov exponents come in symmetric pairs. We define the Fürstenberg group and give some background on Zariski-denseness in order to state the Gol dsheid- Margulis criterion [23]. Borrowing the arguments from [23, 9], we then show for our particular model that the Fürstenberg group is Zariski-dense at all nonzero energies so that by the Gol dsheid-margulis criterion, the Fürstenberg group possesses the desired contractivity and irreducibility properties. Positivity of Lyapunov exponents then follows by a generalization of Fürstenberg s theorem [4]. A separate argument at zero energy shows that Zariski-denseness does not hold at 0, and the first Lyapunov exponent is always positive, while the second may vanish for certain distributions. In Chapter 6 we work in the general setting of ergodic block Jacobi matrices, where both the diagonal and off-diagonal blocks are uniformly bounded and may be made random. We argue existence of the integrated density of states (IDS) and prove a Thouless formula, which allows one to transfer any Hölder continuity from the Lyapunov exponents to the IDS. Such a formula was first introduced for the 1-D Anderson model by Thouless [43] in 1972 and was later made rigorous by Avron and Simon [3]. A Thouless formula was proven for Anderson models on strips by Craig and Simon [12] in 1983, and Kotani and Simon [34] made up a different proof in 1988. We prove ours by generalizing and expanding upon the ideas from [12]. For random block Jacobi matrices with i.i.d. entries, we then prove in Chapter 7 via bootstrap multiscale analysis that subexponential dynamical localization holds only under contractivity and irreducibility assumptions on the Fürstenberg group, which may be checked, for example, by showing Zariski-denseness. In order to obtain the desired localization result, we thus prove an appropriate Wegner estimate and

6 initial length scale estimate [31]. A key input to the Wegner estimate is the Thouless formula we prove. Using a Green function formula for block Jacobi matrices from Appendix B, we then follow the approach in [32] in proving a Wegner estimate and initial length scale estimate, and we conclude localization. Thus, if one is interested in proving localization for a random block Jacobi matrix of this type, one must only verify the assumptions on the Fürstenberg group. Due to the results in Chapter 5, the localization result from Chapter 7 has immediate applicability to the random block operator arising from the xy chain. However, as contractivity and irreducibility cannot be checked at E = 0, we cannot prove localization at all energies, and thus the zero-velocity Lieb-Robinson bound for the anisotropic xy-spin chain in Chapter 2 cannot yet be concluded. One of the main problems, due to the lack of contractivity and irreducibility properties at E = 0, is that one does not automatically have the Hölder continuity of the Lyapunov exponents at 0. Also, there is an issue of non-uniqueness of the invariant measure which one would have to resolve. So an open question arises: can one still prove dynamical localization at all energies without using contractivity and irreducibility at 0, but using that both leading Lyapunov exponents at 0 are positive? Such a critical energy as described above arises already in a simpler model, which we present in Chapter 8. In fact, it corresponds to the random block operator H ω,γ with γ = 1, which degenerates to the Ising model. Here, the transfer matrix formalism does not work, as the hopping terms are no longer invertible. Rather, due to the simple nature, we show how the problem reduces to that of an ordinary Jacobi matrix with off-diagonal randomness. We then prove noncompactness and strong irreducibility

7 of the Fürstenberg group at all nonzero energies, while at zero energy we show that strong irreducibility does not hold. This is of the same flavor as the issue for the original model H ω,γ, but the model is simpler and more conducive to understanding how one might be able to still prove dynamical localization at all energies, allowing one to conclude the desired zero-velocity Lieb-Robinson bound from the results in Chapter 2.

8 CHAPTER 2 QUANTUM SPIN SYSTEMS AND THE MODEL 2.1. The xy-spin Chain Model A major topic in quantum mechanics is the study of many-body systems. In particular, a system of interacting electrons is already a highly nontrivial matter, as the underlying Hilbert space for a single electron is the infinite-dimensional space L 2 (R 3 ). As a first step to understanding interacting electrons, physicists and mathematicians find it much easier to study systems of interacting spins. For our purposes, we will regard a spin ψ = ( ψ + ψ ) as a normalized vector in C 2, normalized because we would like ψ + 2 and ψ 2 to represent the probabilities that the electron has spin up or down, respectively. An axiom of quantum mechanics states that if the underlying Hilbert spaces of two noninteracting particles are H 1 and H 2, then if we let these particles interact, the Hilbert space associated with the pair becomes the tensor product H 1 H 2. So naturally, if we agree to study an xy-spin chain (or just xy chain), which we can visualize as n spins oriented along some line (where typically only next neighbors interact), then the Hilbert space is the n-fold tensor product n C 2. A great deal of the work on the xy chain in recent decades has been motivated by the groundbreaking paper [37] of Lieb, Schultz, and Mattis. In the early 1990s, some rigorous mathematical work was done on the xy chain, being influenced by rapid

progress on the Anderson model and developments of several methods which allow one to rigorously prove localization. For further history and references, we refer to [26]. We now consider the anisotropic xy chain Hamiltonian with free boundary conditions as introduced in [37]: n 1 n H [1,n] = µ j [(1 + γ j )σj x σj+1 x + (1 γ j )σ y j σy j+1 ] + ν j σj z, (2.1) j=1 which acts on the Hilbert space H [1,n] = n C 2. Here {µ j }, {γ j }, and {ν j } are three real-valued sequences, representing the coupling strength, the anisotropy, and the external magnetic field, respectively, and σ x = 0 1 1 0, σy = 0 i i 0, σz = j=1 1 0 0 1 are the Pauli matrices. For a 2 2 matrix M, we use the notation M j := I I M I I, which acts nontrivially only in the jth component. We will abbreviate H n := H [1,n]. To understand the effects of disorder on spin systems, we are particularly interested in the case where the sequences {µ j }, {γ j }, and {ν j } are random. We remark that the first sum in (2.1) models the interactions between neighboring spins, as governed by the x and y Pauli matrices. The second sum models an exterior 9 magnetic field acting on the spin system. We note that the interactions are not uniform along the chain because of the anisotropy parameters γ j. The isotropic case γ j = 0 essentially reduces to the study of the Anderson model and was first rigorously understood in [33] in the context of almost-sure localization of ground state correlations. In the recent paper [26], a more general theorem was proven

10 from which one deduces correlation decay, although in expectation rather than with probability one. As mentioned in [26], it is well known (see, e.g. [37]) that Hamiltonians of the form (2.1) can be diagonalized in terms of a system of operators {b j } satisfying the canonical anticommutation relations (CAR) {b j, b k} = δ jk I, {b j, b k } = {b j, b k} = 0, 1 j, k n, where {A, B} := AB + BA is the anticommutator. We now outline part of this diagonalization procedure. After defining the raising and lowering operators a j = 1 2 (σx j + iσ y j ), a j = 1 2 (σx j iσ y j ), j = 1,..., n, one checks the relations σ x j σ x j+1 + σ y j σy j+1 = 2(a ja j+1 + a j+1a j ) σ x j σ x j+1 σ y j σy j+1 = 2(a j a j+1 + a j+1a j) σ z j = 2a ja j I. In terms of the raising and lowering operators, the operator H n becomes n 1 n H n = 2 µ j [a ja j+1 + a j+1a j + γ j (a j a j+1 + a j+1a j)] + ν j (2a ja j I) (2.2) j=1 j=1 Because these a-operators act locally, they do not satisfy the CAR; for example, for j k, a j and a k commute rather than anticommute. To break this locality, we change variables via the Jordan-Wigner transformation. We set c 1 := a 1, c j := σ z 1 σ z j 1a j, j = 2,..., n. (2.3)

11 Using that {a j, a j } = I, (a j) 2 = 0, and a 2 j = 0, one easily checks that c jc j = a ja j, c jc j+1 = a ja j+1, c j c j+1 = a j a j+1. Rewriting (2.2) in terms of the c-operators, we obtain n 1 n H n = 2 µ j [ c jc j+1 c j+1c j + γ j (c j c j+1 + c j+1c j)] + ν j (2c jc j I). (2.4) j=1 j=1 One may verify that the c-operators do satisfy the CAR, i.e. {c j, c k} = δ jk I, {c j, c k } = {c j, c k} = 0, 1 j, k n. (2.5) We now wish to express the xy Hamiltonian H n in terms of C = (c 1,..., c n, c 1,..., c n) t. By using (2.5), we symmetrize (2.4) to obtain n 1 H n = µ j [ c jc j+1 + c j+1 c j c j+1c j + c j c j+1] j=1 n 1 n + µ j γ j [c j c j+1 c j+1 c j + c j+1c j c jc j+1] + ν j (c jc j c j c j) j=1 j=1 = C ˆMn C, (2.6) where ˆM n is the 2 2 block matrix ˆM n = A n B n B n A n (2.7) with Jacobi matrices A n = ν 1 µ 1 µ 1..................... µ n 1 (2.8) µ n 1 ν n

12 and B n = 0 µ 1 γ 1. µ 1 γ..... 1.......... (2.9)...... µ n 1 γ n 1 µ n 1 γ n 1 0 Observe that A n = A t n = A n and Bn = Bn t = B n so that we have ˆM n = ˆM n t = ˆM n. The authors in [26] then go on to apply a Bogoliubov-type transformation to diagonalize H n in terms of a new system {b j } of operators satisfying the CAR. As a result, the xy chain Hamiltonian takes the form of a free Fermion system, a fact which is exploited in the study the isotropic case. However, we have stopped after introducing the matrix ˆM n because this will be our main object of interest. After applying this Bogoliubov transformation, and switching from the b-operators back to the c-operators, one arrives at the identity n n τt n (c j ) = ˆM n,j,k (2t)c k + ˆM n,j,n+k (2t)c k, j = 1,..., n, (2.10) k=1 k=1 where τ n t (a) := e ithn ae ithn is the Heisenberg dynamics for an operator a on n C 2, and ˆM n,j,k (t) := (e it ˆM n ) j,k is the (j, k)-th matrix element of the time evolution of ˆM n. We will show in the next section that dynamical localization of ˆMn (in some sense) implies a sort of dynamical localization (i.e. a Lieb-Robinson bound) for the anisotropic xy-spin chain. Let us remark about the special structure of ˆMn ; if we apply the change of basis (e 1, e 2,..., e 2n ) (e 1, e n+1, e 2, e n+2,..., e n, e 2n ), we see that ˆM n is unitarily equivalent

to the block matrix ν 1 J µ 1 S(γ 1 ) µ 1 S(γ 1 ) t. ν 2 J.. M n :=, (2.11)...... µ n 1 S(γ n 1 ) µ n 1 S(γ n 1 ) t ν n J where J := 1 0 0 1, S(γ) := 1 γ We may view M n as a generalized tight-binding Hamiltonian with a (sign-indefinite) potential generated by the magnetic field {ν j } and non-standard hopping terms as γ 1. 13 seen on the off-diagonals. Mathematically, (2.11) provides the possibility to investigate spectral properties of ˆMn by using a transfer matrix formalism, although of higher order than in the case of standard tri-diagonal Jacobi matrices. This is the approach we follow in Chapter 5. Let [n 1, n 2 ] := {n 1, n 1 + 1,..., n 2 }. We say that ˆM n is dynamically localized if there exist ζ (0, 1) and constants Ĉ > 0, ˆη > 0 such that for all n N and j, k [1, n], ( ) E sup ˆM n,j,k (t) + sup ˆM n,j,n+k (t) Ĉe ˆη j k ζ. (2.12) t R t R Let P j : l 2 ([1, n]; C 2 ) C 2 be the projection defined by P j u = u(j). We say that M n is dynamically localized if there exist ζ (0, 1) and constants C > 0, η > 0 such that for all n N and j, k [1, n], ( ) E sup P j e itmn Pk t R Ce η j k ζ. (2.13) One can easily see by the unitary equivalence of ˆMn and M n that one is dynamically localized if and only if the other is.

14 Thus, (2.12) and (2.13) are equivalent. The version (2.12) is more convenient to work with while proving a Lieb-Robinson bound in the next section, while the version (2.13) is in a form most convenient to be proven. 2.2. Lieb-Robinson Bound Under the assumption that ˆMn (or M n ) is dynamically localized, in the sense that either (2.12) or (2.13) holds, we will show that the anisotropic xy-spin chain satisfies a zero-velocity Lieb-Robinson bound, which may be thought of as a concept of dynamical localization for the spin chain. This establishes a finite propagation speed for spin waves governed by ˆM n. Understanding whether or not (2.13) holds is thus one of the main goals of this research. A statement such as (2.13), due to the lack of a spectral projection, is a statement about dynamical localization at all energies. However, with the tools we use in Chapters 5 and 7 (e.g. positivity of Lyapunov exponents and multiscale analysis), (2.13) comes out with a spectral projection χ I (M n ) over any interval I not containing zero. The question of whether or not (2.12) or (2.13) holds in general thus remains an open problem in our more long-term research goals. The following is the Lieb-Robinson bound we obtain for the anisotropic xy-spin chain. The notation A N, where N Z, represents the class of local observables which are tensor products of bounded operators acting trivially on the sites outside N. For more background and precise definitions, we refer to [26].

Theorem 2.1. Suppose there exist ζ (0, 1) and constants C > 0, η > 0 such that for all n N and j, k [1, n], ( E sup t R ˆM n,j,k (t) + sup ˆM n,j,n+k (t) t R 15 ) Ce η j k ζ. (2.14) Then for every ε (0, η), there exists a constant C = C (η, ε, ζ) > 0 such that for all 1 j < k and n k ( ) E sup [τt n (A), B] C A B e (η ε)(k j)ζ (2.15) t R for all A A j and B A [k,n]. Proof. Fix 1 j < k, n k, and ε (0, η). Suppose first that A = c j. Then by (2.10), [τ n t (c j ), B] = n k =k ˆM n,j,k (2t)[c k, B] + where we have used B A [k,n] and c k, c k n k =k ˆM n,j,n+k (2t)[c k, B], A [1,k ] so that [c k, B] = [c k, B] = 0 for k < k. By (2.14) and the fact that [S, T ] 2 S T, ( ) n E sup [τt n (c j ), B] 2C B e η(k j) ζ. (2.16) t R Here, we must use an argument different from that in [26] since this series is not geometric. We have Substituting we have k j 1 n e η(k j) ζ = k =k e ηxζ dx = n j m=k j e ηmζ l=k j k =k e ηmζ u = x ζ du = ζ dx = ζ x1 ζ ζ 1 u 1 ζ ζ (k j 1) ζ u 1 ζ ζ e ηu du C 1 k j 1 dx, e ηxζ dx. (2.17) (k j 1) ζ e (η ε/2)u du (2.18)

16 where C 1 = C 1 (ε, ζ) > 0 is chosen large enough so that ζu 1 ζ ζ e ηu C 1 e (η ε/2)u for all u 0. Now (k j 1) ζ e (η ε/2)u du = 1 η ε/2 e (η ε/2)(k j 1)ζ where we have used the fact that x ζ (x 1) ζ + 1 for all x 1. eη ε/2 η ε/2 e (η ε/2)(k j)ζ, (2.19) Putting together Equations (2.16) (2.19), there is a constant C 2 = C 2 (ε, ζ, η, C) > 0 such that ( ) E sup [τt n (c j ), B] t R C 2 B e (η ε/2)(k j)ζ, (2.20) an analogue of Equation (3.20) in [26]. By taking adjoints, the case A = c j follows with the identical estimate ( ) E sup [τt n (c j), B] C 2 B e (η ε/2)(k j)ζ, (2.21) t R Now suppose that A = a j. By (2.3), we have a j = σ z 1 σ z j 1c j. Using the automorphism property τ n t (ST ) = τ n t (S)τ n t (T ) and the Leibnitz rule for commutators [ST, U] = S[T, U] + [S, U]T, we have that [τ n t (a j ), B] = τ n t (σ z 1) τ n t (σ z j 1)[τ n t (c j ), B] + [τ n t (σ z 1) τ n t (σ z j 1), B]τ n t (c j ). From this we conclude ( ) E sup [τt n (a j ), B] t R ( C 2 B e (η ε/2)(k j)ζ + E sup [τt n (σ1) z τt n (σj 1), z B] t R where we have used (2.20). We will now work on the second term above. For 1 l < k, it is convenient to define ( ) C(l, B) := E sup [τt n (σ1) z τt n (σl z ), B] t R and C(0, B) := 0, ), (2.22)

17 and we claim that C(l, B) C(l 1, B) + 4C 2 B e (η ε/2)(k l)ζ l = 1,..., k 1. (2.23) To see this, first note that with another application of the Leibnitz rule, we have [τ n t (σ z 1) τ n t (σ z l ), B] [τ n t (σ z l ), B] + [τ n t (σ z 1) τ n t (σ z l 1), B], (2.24) where the second term on the right-hand side is 0 if l = 1. Furthermore, using that σ z l = 2a la l I = 2c lc l I l, we obtain and thus using (2.20) and (2.21), [τ n t (σ z l ), B] = 2[τ n t (c l), B]τ n t (c l ) + 2τ n t (c l)[τ n t (c l ), B] [τ n t (σ z l ), B] 4C 2 B e (η ε/2)(k l)ζ l. This estimate, in conjunction with (2.24), proves (2.23). Our desired estimate follows by iteration. From (2.22) and (2.23), we have ( ) E sup [τt n (a j ), B] t R C 2 B e (η ε/2)(k j)ζ + C(j 1, B) j 4C 2 B e (η ε/2)(k l)ζ 4C 2 B l=1 m=k j e (η ε/2)mζ C 3 B e (η ε)(k j)ζ (2.25) for some new constant C 3 = C 3 (ε, ζ, η, C) > 0, where, in the last inequality above, we have repeated the argument from Equations (2.17) (2.19). Again by taking adjoints, the case A = a j follows with the same estimate as (2.25). Using the Leibnitz formula, the cases A = a ja j and A = a j a j follow with

18 the same estimate but with an extra factor 2. Since the collection { } a j, a j, a ja j, a j a j constitutes the canonical basis for A j, we obtain (2.15) with C := 6C 3. We end this chapter with some remarks. (i) By the results from Chapters 5 and 7, we fall just short of obtaining (2.14) as contractivity and irreducibility cannot be checked at E = 0. Thus, the Lieb- Robinson bound for the anisotropic xy-spin chain may not yet be concluded. This poses the interesting question of whether dynamical localization at all energies can be salvaged in the case that both leading Lyapunov exponents at 0 are positive, using that the more powerful machinery of contractivity and irreducibility hold at all nonzero energies. (ii) The proof of Theorem 2.1 mimics that of Theorem 3.2 in [26]. However, in [26], some smoothness of the single-site distribution was needed in order to conclude (2.14) with ζ = 1 for the isotropic xy-spin chain via the Kunz- Souillard method or the fractional moment method. Such a statement with ζ = 1 is stronger and thus yields the stronger Lieb-Robinson bound as in [26]. However, since (2.14) holds by multiscale analysis [21, 31] for the isotropic xyspin chain with ζ (0, 1) but for arbitrary nontrivial distributions, Theorem 2.1 may also be applied in this case to obtain a Lieb-Robinson bound. (iii) In [16], the authors adapt the fractional moment method and prove dynamical localization for a more general class of models in the large disorder regime, but again using smoothness of the distribution. As a result, (2.14) with ζ = 1 holds for the anisotropic xy-spin chain, but with randomness only in the potential. Thus, for large disorder, one can conclude (2.15) with ε = 0 and ζ = 1.

19 2.3. Emergence of the Random Block Operator When we introduced the Hamiltonian H [1,n] in the beginning of Section 2.1, we defined three random sequences: the couplings {µ j }, the anisotropy factors {γ j }, and the external magnetic field strengths {ν j }. To understand the effects of adding such randomness (in particular, how to prove (2.13)), it is logical and easier to randomize one quantity at a time, letting the probability distributions of the other two be concentrated on a single point. To this end, let us assume µ j = 1 and γ j = γ [0, 1] for all j, while we randomize the ν j. Let us also rename ω j := ν j to our favorite symbol denoting randomness. We thus have where V ω (1) S(γ) S(γ) t. V ω (2).. H ω,γ,n := M n =........., (2.26)...... S(γ) S(γ) t V ω (n) V ω (j) := ω j 0 0 ω j, S(γ) := 1 γ In addition to studying this finite-volume operator, let us also extend the block matrix (2.7) to the infinite-volume block operator Ĥ ω,γ = A ω γb γb A ω γ 1., (2.27) which acts on the Hilbert space l 2 (Z) l 2 (Z). Here ω = (ω j ) j Z is taken from a probability space (Ω, F, P) = j Z(R, B(R), µ),

20 where we assume that the probability measure µ is nontrivial and compactly supported, i.e. supp(µ) contains at least two elements and is compact. The operators A ω and B defined by (A ω u)(n) = u(n + 1) u(n 1) + ω n u(n) (Bu)(n) = u(n 1) u(n + 1), for n Z and u l 2 (Z), are thus bounded operators on l 2 (Z). If we define the discrete Laplacian h 0 by (h 0 u)(n) = u(n + 1) + u(n 1), we see that A ω = h 0 + V ω is the Anderson model (which is self-adjoint), and B is a skew-symmetric Laplaciantype operator with non-standard hopping terms. It follows that Ĥω,γ is bounded and self-adjoint on l 2 (Z) l 2 (Z). (Boundedness, in fact, will follow from Proposition 3.2.) This operator is interesting, both from a purely mathematical perspective and from the viewpoint of quantum spin systems, when one might be interested in studying the behavior of a spin chain in the thermodynamic limit, i.e. as its length n goes to infinity. One runs into trouble when considering an infinite spin system because infinite tensor products may not be defined, which is why one considers finite spin systems, tries to prove bounds that are uniform in the volume, and then considers the thermodynamic limit in order to understand qualitatively how an infinite system would behave. Moreover, as explained in the first remark following Theorem 7.1, the methods we have chosen for proving localization in Chapter 7 yield localization for both the finite-volume and infinite-volume operators simultaneously. Thus, we will be content to study the infinite-volume operator (2.27) and its unitary equivalent (2.28) because of their readiness for implementing a transfer matrix formalism.

In the isotropic case γ = 0, the block operator (2.27) is simply a direct sum A ω ( A ω ), whose study is trivially connected with that of the Anderson model. However, in the anisotropic case γ (0, 1), the off-diagonal terms couple the two Anderson models, and the study quickly becomes nontrivial. Until Chapter 8, we will avoid the case γ = 1, because the model degenerates into the Ising model, which necessitates that one uses different strategies since the matrix S(1) is not invertible. Analogously to the finite-volume case, one may define the unitary transformation U : l 2 (Z) l 2 (Z) l 2 (Z; C 2 ) by ( ) ( ) ϕ + (n) ϕ + (Uϕ)(n) =, ϕ = l 2 (Z) l 2 (Z), n Z. ϕ (n) ϕ 21 One can verify that UĤω,γU = H ω,γ, where......... V ω ( 1) S(γ) H ω,γ := S(γ) t V ω (0) S(γ) S(γ) t. V ω (1)........, (2.28) where V ω (n) := ω n 0 0 ω n and S(γ) := 1 γ In Chapter 5, we will apply transfer matrix methods to answer questions such as positivity of the Lyapunov exponents, which will be an input to the proof of localization. γ 1.

22 CHAPTER 3 BASIC SPECTRAL PROPERTIES This chapter establishes various spectral properties of a more general class of block operators, of which (2.27) is a special case. The proofs of these properties are adapted from a similar class of block operators considered in [30] which arise in the study of dirty semiconductors. In particular, we will infer boundedness of the operator H ω,γ and establish a condition under which H ω,γ has a spectral gap. Let H be a Hilbert space, and let A B(H) be self-adjoint and B B(H) be skew-adjoint (i.e. B = B). Then the block operator H := A B B A (3.1) is self-adjoint on the Hilbert space H H, as one can verify it is Hermitian. The inner product on H H is given by ( ) ( ) f + g +, f g H H := f +, g + H + f, g H. We then have the following properties regarding the spectrum of H: Lemma 3.1. (i) The spectrum of H is symmetric about 0, i.e. σ(h) = σ(h). (ii) If Hϕ = Eϕ for some ϕ = ( ϕ + ϕ ) H H, then H ϕ = E ϕ, where ϕ = ( ϕ ϕ + ).

23 Proof. For part (i), define the unitary block operator U 1 := 0 I I 0 Then one calculates U 1HU 1 = H, which proves symmetry of the spectrum. For part (ii), H ϕ = HU 1 ϕ = U 1 Hϕ = U 1 (Eϕ) = E ϕ.. Proposition 3.2. With A, B, and H as above, (i) σ(h) [ A B, A + B ]; (ii) If there exists a λ 0 such that A λ or A λ, then σ(h) ( λ, λ) =. (3.2) Proof. For part (i), first note that A 0 A, 0 A for we have A 0 0 A Similarly we have Thus, A B ( ϕ+ ϕ ) 2 ( ) = Aϕ+ 2 Aϕ 0 B B. B 0 ( ) = Aϕ + 2 + Aϕ 2 A 2 ϕ+ 2. ϕ B A 0 0 B + A + B A 0 A B 0 so that σ(h) [ H, H ] [ A B, A + B ].

For part (ii), assume A λ. (The case A λ is similar.) If we define the 24 unitary block operator then one calculates U 2H 2 U 2 = U 2 U 2 := 1 2 I I A 2 B 2 [A, B] [A, B] A 2 B 2 I I, U 2 = K + 0 0 K, where K ± := A 2 B 2 ± [A, B], and [A, B] := AB BA is the commutator. Defining à := A λ 0 and using that à is self-adjoint and B skew-adjoint, it follows that K ± = (à + λ)2 B 2 ± [à + λ, B] = Ã2 B 2 ± [Ã, B] + 2λà + λ2 = (à B)(à ± B) + 2λà + λ2 = (à ± B) (à ± B) + 2λà + λ2 0 + 0 + λ 2 = λ 2. This implies σ(h 2 ) = σ(k + ) σ(k ) [λ 2, ). Now if s σ(h), then s 2 σ(h 2 ) so that s 2 λ 2. Thus, s (, λ] [λ, ), i.e. σ(h) ( λ, λ) =.

25 CHAPTER 4 ALMOST-SURE SPECTRUM 4.1. The Anderson Model Consider the one-dimensional Anderson model h ω = h 0 +V ω. Because this family of operators is ergodic, one concludes from Pastur s theorem [14] that the spectrum of h ω is almost surely deterministic, i.e. that there is a closed set Σ R such that σ(h ω ) = Σ almost surely. For general ergodic operators, one might not know any more than the fact that Σ exists. However, for the Anderson model, an explicit formula is known, namely Σ = [ 2, 2] + supp(µ). (4.1) Another way of stating this result is the following: for c supp(µ), let h c = h 0 +V c, where V c = (..., c, c, c,...) is a constant potential. Then the almost-sure spectrum is the union of the spectra over all constant potentials taking values within the range of the random parameters. To see this, note that, since σ(h 0 ) = [ 2, 2] and constant potentials merely shift the spectrum, we may write Σ = [ 2, 2] + supp(µ) = = ([ 2, 2] + c) c supp(µ) c supp(µ) σ(h c ). (4.2)

Thus, one observes that constant potentials are enough to generate the entire almostsure spectrum of the Anderson model. 26 4.2. A Periodic Support Theorem Consider now the random block operator of interest: H ω,γ := = A ω γb h 0 γb γb h 0 γb A ω + V ω 0 0 V ω =: H 0,γ + V ω. (4.3) Above, recall that A ω is the Anderson model. The simple structure of H ω,γ along with ergodicity of the Anderson model implies ergodicity of H ω,γ. Thus, we are assured of the existence of a closed set Σ γ R such that σ(h ω,γ ) = Σ γ almost surely. The natural question to ask now is the following: due to the simple structure of H ω,γ and the explicit formula (4.1) for the almost sure spectrum of the Anderson model, can we determine an explicit formula for Σ γ? It turns out that a formula as nice as (4.1) does not seem possible to obtain because, as we will see, constant potentials will not always be enough to generate the entire spectrum of H ω,γ. However, as a good first step to understanding the spectrum of H ω,γ, it will be nice to know that, while constant potentials may or may not be enough, periodic potentials are always enough. Theorems of this type are referred to as periodic support theorems and seem to have made their first appearance in [28]. To this end, let S per := {V : Z C : V periodic, V (n) supp(µ) n Z}

27 be the space of all periodic sequences taking values in supp(µ). For an element ϕ = ( ϕ + ϕ ) l 2 (Z) l 2 (Z), we define supp(ϕ) to be the set of those n Z for which not both ϕ + (n) and ϕ (n) are zero. Note: this definition is more obvious if we identify l 2 (Z) l 2 (Z) = l 2 (Z; C 2 ). Then supp(ϕ) is the set of those n Z for which ϕ(n) = ( ϕ + (n) ϕ (n)) ( 0 0). Also, for the remainder of this section, we will not carry around the dependence of the model on γ, so we write Σ = Σ γ, H ω = H ω,γ, and H 0 = H 0,γ. Theorem 4.1 (Periodic Support Theorem). We have Σ = V S per σ(h V ) =: Σ 0. (4.4) Proof. ( ) It suffices to show that σ(h ω ) Σ 0 for every ω Ω. So fix an ω Ω and an E σ(h ω ). This implies the existence of a Weyl sequence (ϕ n ) n N l 2 (Z) l 2 (Z), i.e. ϕ n = 1 and (H ω E)ϕ n 0. Without loss of generality, we may assume that these ϕ n have finite support, say supp(ϕ n ) [ K n, K n ], since these finitely supported sequences form a core for any operator defined on all of l 2 (Z) l 2 (Z). For n N, define V (n) ω : l 2 (Z) l 2 (Z) and V (n) ω : l 2 (Z) l 2 (Z) l 2 (Z) l 2 (Z) by (V (n) ω u)(m) := V (n) ω := ω m u(m) V (n) ω 0 0 V (n) ω if 2K n < m 2K n 4K n -periodically extended.

28 Making the natural definition H (n) ω := H 0 + V (n) ω, which is a periodic operator, we notice that H (n) ω ϕ n = H ω ϕ n. Thus, (H (n) ω E)ϕ n = (H ω E)ϕ n =: ε n 0. (4.5) It is a general fact that for self-adjoint T, f D(T ), and E C, that (T E)f dist(σ(t ), E) f. Hence, using that ϕ n = 1, (4.5) implies Since we know that σ(h (n) ω ) Σ 0, then dist(σ(h (n) ω ), E) ε n. dist(σ 0, E) dist(σ(h (n) ω ), E) ε n 0 so that dist(σ 0, E) = 0. Now Σ 0 is closed, so the distance is realized by a point in Σ 0, which shows that E Σ 0. ( ) This inclusion will be proven in a couple of steps. (i) We first show that, for fixed V S per, we have σ(h V ) σ(h ω ) for a.e. ω Ω. Fix N N and ε > 0, and let Ω N,ε := {ω Ω : k Z : ω n V (n) < ε n [k, k + N)}. For n Z, let p n := µ((v (n) ε, V (n) + ε)) > 0 (since V (n) supp(µ)). Because V is periodic, then there are finitely many distinct values of p n. Let p be the minimum of these; note p > 0. Then, for every k Z, P({ω Ω : ω n V (n) < ε n [k, k + N)}) = k+n 1 n=k p n p N > 0. (4.6) Since we may cover Z by countably many disjoint intervals of the form [k, k + N), the corresponding sets appearing inside the measure in (4.6) are all independent and

29 have positive measure. Thus, we have P(Ω N,ε ) = 1 (a countable union of independent events with uniformly positive measure has full measure). Now if we let Ω ε := Ω N,ε, Ω 0 := Ω 1/l, N N l N then P(Ω ε ) = 1 for all ε > 0, and P(Ω 0 ) = 1. We claim that for all ω Ω ε and all E σ(h V ), it is true that [E ε, E + ε] σ(h ω ). (4.7) To show this, since E σ(h V ), we deduce from Theorem A.14 that there exists a θ [0, 2π) such that E σ(h L,θ ) (here, L is the period of V ), where H L,θ := H L,θ γb L,θ γb L,θ H L,θ, (4.8) and H L,θ = B L,θ = V (1) 1 0 e iθ 1 V (2) 1 0.. 1......, (4.9). 0... V (L 1) 1 e iθ 0 1 V (L) 0 1 0 e iθ 1 0 1 0.. 1....... (4.10). 0... 0 1 e iθ 0 1 0

Thus we are guaranteed a corresponding eigenvector ϕ, i.e. H L,θ ϕ = Eϕ. If we extend ϕ to a function on Z via ϕ(n + L) = e iθ ϕ(n) for every n Z, then ϕ is a generalized 30 eigenfunction of H V to E, i.e. it is not in l 2 (Z; C 2 ), but it solves the finite difference equation H V ϕ = Eϕ. Since ω Ω ε, then for every N N, there exists a k N Z such that ω n V (n) < ε for all n Λ N := [k N, k N + N). Then ψ N := (χ ΛN χ ΛN )ϕ has finite support and is, in particular, in l 2 (Z) l 2 (Z). We claim that lim sup N (H ω E)ψ N ψ N ε. (4.11) To show (4.11), note that (H ω E)ψ N = (H ω H V )ψ N + (H V E)ψ N = (V ω V)ψ N + (H V E)ψ N Because ϕ solves the finite difference equation H V ϕ = Eϕ, then ((H V E)ψ N )(n) is nonzero only for n close to the boundary of Λ N, i.e. for n = k N 1, k N, k N + N 1, k N + N, and its values for these n are bounded by a constant independent of N. So we have (H ω E)ψ N (V ω V)ψ N + (H V E)ψ N ε ψ N + C (4.12) Dividing (4.12) by ψ N, letting N, and noting that ϕ / l 2 (Z; C 2 ) implies ψ N as N, we arrive at (4.11).

Now to conclude (4.7): if H ω E is not injective, then E σ(h ω ) and (4.7) holds. If H ω E is injective, then (4.11) implies 31 (H ω E) 1 1 ε. (4.13) Since H ω is self-adjoint, then (H ω E) 1 = 1 dist(σ(h ω ), E). (4.14) Equations (4.13) and (4.14) imply that dist(σ(h ω ), E)) ε, which establishes (4.7). Now (4.7) implies that for ω Ω 0, and E σ(h V ), we have for every l N, [E 1/l, E + 1/l] σ(h ω ). Because σ(h ω ) is closed, we have E σ(h ω ). We have shown that σ(h V ) σ(h ω ) for all ω Ω 0, which completes part (i). (ii) We wish to show that Σ 0 Σ, given that for every V S per, σ(h V ) σ(h ω ) for all ω Ω V, where P(Ω V ) = 1. We must construct a countable subset S per S per, enumerated as S per = {V (j) } j N, such that σ(h V ) = σ(h V (j)). (4.15) V S per j N We would then be done, for we would have for every ω Ω 0 := j N Ω V (j), Σ 0 σ(h ω ), from which we can conclude that Σ 0 Σ. Define the countable subset to be S per := {V : Z C : V periodic, V (n) supp(µ) Q n Z}. To prove (4.15), It is sufficient to show that for every V S per, we have σ(h V ) j N σ(h V (j)). Suppose V has period L, and fix E σ(h V ). Then by Theorem A.14, there is a θ [0, 2π) such that E is an eigenvalue of (4.8), where (4.8) is defined

32 by (4.9) and (4.10). By approximating each of V (1),..., V (L) (appearing in (4.9)) closer and closer by rational numbers, then because eigenvalues depend continuously on matrix elements, we can obtain a sequence {E n } n N such that E n E and E n is an eigenvalue of H V (jn) for some j n N. Then since each E n j N σ(h V (j)), which is closed, we have E j N σ(h V (j)). 4.3. Inclusions for the Spectrum of H ω,γ In this section, we would like to find formulas (or at least set inclusions) for the almost-sure spectrum of H ω,γ. For the Anderson model, as seen in Section 4.1, the almost-sure spectrum is seen to be simply a union over all spectra of operators with constant potentials. For the block operator case, the coupling operator B makes matters not so straightforward, and we show that a formula as nice as (4.1) is not possible for H ω,γ. As a first step to see this, we consider block operators with constant potentials. Their spectra are easy to calculate via the Fourier transform. For c supp(µ), let us write H c,γ = A c γb γb A c = h 0 + c γb γb h 0 c i.e. H c,γ is the block operator H ω,γ with constant potential c. We have the following characterization of the spectrum of H c,γ :,

Lemma 4.2. Let c R and γ 0. Then there exists a closed interval I c,γ [0, ) 33 such that σ(h c,γ ) = ( I c,γ ) I c,γ. We have and 2 + c if 0 γ max(i c,γ ) = γ 4 + c2 if γ γ 2 1 1 + c 2 1 + c 2 (4.16) c 2 if c 2 min(i c,γ ) = 2 c if c 2, γ 1 c 2 (4.17) γ 4 c2 if c 2, 0 γ 1 c. 1 γ 2 2 Before we prove this lemma, we remark that one must find it reasonable that both the interval I c,γ and its negative appear because of the symmetry of the spectrum about 0 which we saw from Lemma 3.1(i). Proof. With the Fourier transform F : L 2 ([0, 2π)) l 2 (Z) given by (F g)(n) := 1 2π 2π 0 g(x)e ixn dx, which is unitary, the following calculation shows that F h 0 F = 2 cos( ), as a multiplication operator on L 2 ([0, 2π)): for g L 2 ([0, 2π)) and n Z, (h 0 F g)(n) = (F g)(n 1) + (F g)(n + 1) = = 1 2π 2π 0 1 2π 2π 0 g(x)(e ix(n 1) + e ix(n+1) )dx 2 cos(x)g(x)e ixn dx = (F (2 cos( )g))(n).

34 A similar calculation shows that F BF = 2i sin( ). Thus, we obtain the unitary equivalence F 0 0 F H c,γ F 0 0 F = F h 0 F + c γf BF γf BF F h 0 F c = 2 cos( ) + c 2iγ sin( ) 2iγ sin( ) 2 cos( ) c =: M c,γ which is a 2 2-block multiplication operator on L 2 ([0, 2π)) L 2 ([0, 2π)). If we let λ ± (x) denote the two eigenvalues of the 2 2-matrix 2 cos(x) + c 2iγ sin(x) 2iγ sin(x) 2 cos(x) c, then one easily modifies the proof of Lemma A.7 to conclude that σ(m c,γ ) = x [0,2π) Now one calculates that these eigenvalues are {λ + (x), λ (x)}. λ ± (x) = ± (2 cos(x) c) 2 + 4γ 2 sin 2 (x). Since λ + : [0, 2π] [0, ) is continuous, its range is a closed interval in [0, ). Thus, as λ = λ +, we obtain that σ(h c,γ ) = ( I c,γ ) I c,γ, where { } I c,γ := (2 cos(x) c) 2 + 4γ 2 sin 2 (x) : x [0, 2π]. The remaining objective is to find expressions for the maximum and minimum of this interval I c,γ. Thus, we must maximize and minimize the function g := λ 2 + over the interval [0, 2π]. Geometrically, we must maximize and minimize the distance

35 (2 cos(x) c)2 + 4γ 2 sin 2 (x) from the origin to the point (2 cos(x) c, 2γ sin(x)) on an ellipse centered at ( c, 0) with x-radius 2 and y-radius 2γ. 1. First, let us consider the case c > 2, the geometry of which is depicted in Figure Figure 1. Geometry behind σ(h c,γ ) for c > 2 and small γ Since the ellipse lies in the left half-plane, it is immediately obvious that min(i c,γ ) = 2 c = c 2. However, the maximum value is not so clear. If γ is sufficiently small, possibly as depicted in Figure 1, then the maximum is max(i c,γ ) = 2 c = 2 + c because all distances to points on the ellipse stay within a circle of radius 2 + c. However, if γ is sufficiently large, as depicted in Figure 2, then the maximum is larger than 2 + c and is highly dependent on γ. These observations already show that the formula we are trying to prove is reasonable. We calculate g (x) = 4 sin(x)[2(γ 2 1) cos(x) + c],