Harper Operators Research Articles

On the role of symmetries in the theory of photonic crystals

We discuss the role of the symmetries in photonic crystals and classify them according to the Cartan–Altland–Zirnbauer scheme. Of particular importance are complex conjugation C and time-reversal T, but we identify also other significant symmetries. Borrowing the jargon of the classification theory of topological insulators, we show that C is a “particle–hole-type symmetry” rather than a “time-reversal symmetry” if one considers the Maxwell operator in the first-order formalism where the dynamical Maxwell equations can be rewritten as a Schrödinger equation; The symmetry which implements physical time-reversal is a “chiral-type symmetry”. We justify by an analysis of the band structure why the first-order formalism seems to be more advantageous than the second-order formalism. Moreover, based on the Schrödinger formalism, we introduce a class of effective (tight-binding) models called Maxwell–Harper operators. Some considerations about the breaking of the “particle–hole-type symmetry” in the case of gyrotropic crystals are added at the end of this paper.

The algebraic geometry of Harper operators

Following an approach developed by Gieseker, Knörrer and Trubowitz for discretized Schrödinger operators, we study the spectral theory of Harper operators in dimensions 2 and 1, as a discretized model of magnetic Laplacians, from the point of view of algebraic geometry. We describe the geometry of an associated family of Bloch varieties and compute their density of states. Finally, we also compute some spectral functions based on the density of states. We discuss the difference between the cases with rational or irrational parameters: for the two-dimensional Harper operator, the compactification of the Bloch variety is an ordinary variety in the rational case and an ind-pro-variety in the irrational case. This gives rise, at the algebro-geometric level of Bloch varieties, to a phenomenon similar to the Hofstadter butterfly in the spectral theory. In dimension 2, the density of states can be expressed in terms of period integrals over Fermi curves, where the resulting elliptic integrals are independent of the parameters. In dimension 1, for the almost Mathieu operator, with a similar argument, we find the usual dependence of the spectral density on the parameter, which gives rise to the well-known Hofstadter butterfly picture.

Spectral relationships between kicked Harper and on-resonance double kicked rotor operators

Kicked Harper operators and on-resonance double kicked rotor operators model quantum systems whose semiclassical limits exhibit chaotic dynamics. Recent computational studies indicate a striking resemblance between the spectra of these operators. In this paper we apply C∗-algebra methods to explain this resemblance. We show that each pair of corresponding operators belongs to a common rotation C∗-algebra Bα, prove that their spectra are equal if α is irrational, and prove that the Hausdorff distance between their spectra converges to zero as q increases if α=p/q with p and q coprime integers. Moreover, we show that corresponding operators in Bα are homomorphic images of mother operators in the universal rotation C∗-algebra Aα that are unitarily equivalent and hence have identical spectra. These results extend analogous results for almost Mathieu operators. We also utilize the C∗-algebraic framework to develop efficient algorithms to compute the spectra of these mother operators for rational α and present preliminary numerical results that support the conjecture that their spectra are Cantor sets if α is irrational. This conjecture for almost Mathieu operators, called the ten Martini problem, was recently proven after intensive efforts over several decades. This proof for the almost Mathieu operators utilized transfer matrix methods, which do not exist for the kicked operators. We outline a strategy, based on a special property of loop groups of semisimple Lie groups, to prove this conjecture for the kicked operators.

Approximating spectral invariants of Harper operators on graphs II

We study Harper operators and the closely related discrete magnetic Laplacians (DML) on a graph with a free action of a discrete group, as defined by Sunada. The spectral density function of the DML is defined using the von Neumann trace associated with the free action of a discrete group on a graph. The main result in this paper states that when the group is amenable, the spectral density function is equal to the integrated density of states of the DML that is defined using either Dirichlet or Neumann boundary conditions. This establishes the main conjecture in a paper by Mathai and Yates. The result is generalized to other self adjoint operators with finite propagation speed.

Approximating Spectral Invariants of Harper Operators on Graphs

We study Harper operators and the closely related discrete magnetic Laplacians (DML) on a graph with a free action of a discrete group, as defined by Sunada (Sun). A main result in this paper is that the spectral density function of DMLs associated to rational weight functions on graphs with a free action of an amenable discrete group can be approximated by the average spectral density function of the DMLs on a regular exhaustion, with either Dirichlet or Neumann boundary conditions. This then gives a criterion for the existence of gaps in the spectrum of the DML, as well as other interesting spectral properties of such DMLs. The technique used incorporates some results of algebraic number theory.

Semi-classical limit for random walks

Let ( G , μ ) (G, \mu ) be a discrete symmetric random walk on a compact Lie group G G with step distribution μ \mu and let T μ T_{\mu } be the associated transition operator on L 2 ( G ) L^2(G) . The irreducibles V ρ V_{\rho } of the left regular representation of G G on L 2 ( G ) L^2(G) are finite dimensional invariant subspaces for T μ T_{\mu } and the spectrum of T μ T_{\mu } is the union of the sub-spectra σ ( T μ ↿ V ρ ) \sigma (T_{\mu }\upharpoonleft _{V_{\rho }}) on the irreducibles, which consist of real eigenvalues { λ ρ 1 , . . . , λ ρ dim ⁡ V ρ } \{ \lambda _{\rho 1},...,\lambda _{\rho \dim V_{\rho }}\} . Our main result is an asymptotic expansion for the spectral measures \[ m ρ μ ( λ ) := 1 dim ⁡ V ρ ∑ j = 1 dim ⁡ V ρ δ ( λ − λ ρ j ) m_{\rho }^{\mu }(\lambda ) := \frac {1}{\dim V_{\rho }} \sum _{j=1}^{\dim V_{\rho }} \delta (\lambda - \lambda _{\rho j}) \] along rays of representations in a positive Weyl chamber t + ∗ \mathbf {t}^*_+ , i.e. for sequences of representations k ρ k \rho , k ∈ N k\in \mathbb {N} with k → ∞ k\rightarrow \infty . As a corollary we obtain some estimates on the spectral radius of the random walk. We also analyse the fine structure of the spectrum for certain random walks on U ( n ) U(n) (for which T μ T_{\mu } is essentially a direct sum of Harper operators).

Semiclassical study of the spectrum of the sum of two Harper's operators

We give a semiclassical description of the spectrum of an operator equal to the direct sum of two Harper's operators. We localise the part of the spectrum in [2 + ɛ 0, 4], with ɛ 0 > 0, in exponentially small intervals. Then we prove that the Lebesgue measure of the spectrum in the rational case is bounded from below by a strictly positive constant.