arXiv subjects
Publications and source records attributed to David Damanik.
We show that there exist limit-periodic Schrödinger operators such that the associated integrated density of states is Lipschitz continuous. These operators arise in the inverse spectral theoretic KAM approach of Pöschel.
We study an initial value problem for the Toda lattice with almost periodic initial data. We consider initial data for which the associated Jacobi operator is absolutely continuous and has a spectrum satisfying a Craig-type condition, and show the boundedness and almost periodicity in time and space of solutions.
We provide a complete and self-contained proof of spectral and dynamical localization for the one-dimensional Anderson model, starting from the positivity of the Lyapunov exponent provided by Fürstenberg's theorem. That is, a Schrödinger operator in $\ell^2(\mathbb{Z})$ whose potential is given by independent identically distributed (i.i.d.) random variables almost surely has pure point spectrum with exponentially decaying eigenfunctions and its unitary group exhibits exponential off-diagonal decay, uniformly in time. This is achieved by way of a new result: for the Anderson model, one typically has Lyapunov behavior for all generalized eigenfunctions. We also explain how to obtain analogous statements for extended CMV matrices whose Verblunsky coefficients are i.i.d., as well as for half-line analogs of these models.
We provide an elementary proof of the equivalence of various notions of uniform hyperbolicity for a class of $\mathrm{GL}(2,\mathbb{C})$ cocycles and establish a Johnson-type theorem for extended CMV matrices, relating the spectrum to the points on the unit circle for which the associated Szegő cocycle is not uniformly hyperbolic.
We study the quasi-periodic Schrödinger operator $$ -ψ"(x) + V(x) ψ(x) = E ψ(x), \qquad x \in \mathbb{R} $$ in the regime of "small" $V(x) = \sum_{m\in\mathbb{Z}^ν}c(m)\exp (2πi mωx)$, $ω= (ω_1, \dots, ω_ν) \in \mathbb{R}^ν$, $|c(m)| \le \varepsilon \exp(-κ_0|m|)$. We show that the set of reflectionless potentials isospectral with $V$ is homeomorphic to a torus. Moreover, we prove that any reflectionless potential $Q$ isospectral with $V$ has the form $Q (x) = \sum_{m \in \mathbb{Z}^ν} d(m) \exp (2πi mωx)$, with the same $ω$ and with $|d(m)| \le \sqrt{2 \varepsilon} \exp(-\frac{κ_0}{2} |m|)$. Our derivation relies on the study of the approximation via Hill operators with potentials $\tilde V (x) = \sum_{m \in \mathbb{Z}^ν} c(m) \exp (2 πi m \tilde ωx)$, where $\tilde ω$ is a rational approximation of $ω$. It turns out that the multi-scale analysis method of \cite{DG} applies to these Hill operators. Namely, in \cite{DGL} we developed the multi-scale analysis for the operators dual to the Hill operators in question. The main estimates obtained in \cite{DGL} allow us here to establish the estimates for the gap lengths and the Fourier coefficients in a form which is considerably stronger than the estimates known in the theory of Hill operators with analytic potentials in the general setting. Due to these estimates, the approximation procedure for the quasi-periodic potentials is effective, despite the fact that the rate of approximation $|ω- \tilde ω| \thicksim \tilde T^{-δ}$, $0 < δ< 1/2$ is slow, on the scale of the period $\tilde T$ of the Hill operator.
We study the dynamics given by the iteration of a (half-line) CMV matrix with sparse, high barriers. Using an approach of Tcheremchantsev, we are able to explicitly compute the transport exponents for this model in terms of the given parameters. In light of the connection between CMV matrices and quantum walks on the half-line due to Cantero-Grünbaum-Moral-Velázquez, our result also allows us to compute transport exponents corresponding to a quantum walk which is sparsely populated with strong reflectors. To the best of our knowledge, this provides the first rigorous example of a quantum walk which exhibits quantum intermittency, i.e., nonconstancy of the transport exponents. When combined with the CMV version of the Jitomirskaya-Last theory of subordinacy and the general discrete-time dynamical bounds from Damanik-Fillman-Vance, we are able to exactly compute the Hausdorff dimension of the associated spectral measure.
We rigorously prove a new kind of anomalous (or sub-ballistic) Lieb-Robinson bound for the isotropic XY chain with Fibonacci external magnetic field at arbitrary coupling. It is anomalous in that the usual exponential decay in $x-vt$ is replaced by exponential decay in $x-vt^α$ with $0<α<1$. In fact, we can characterize the values of $α$ for which such a bound holds as those exceeding $α_u^+$, the upper transport exponent of the one-body Fibonacci Hamiltonian. Following the approach of \cite{HSS11}, we relate Lieb-Robinson bounds to dynamical bounds for the one-body Hamiltonian corresponding to the XY chain via the Jordan-Wigner transformation; in our case the one-body Hamiltonian with Fibonacci potential. We can bound its dynamics by adapting techniques developed in \cite{DT07, DT08, D05, DGY} to our purposes. We also explain why our method does not extend to yield anomalous Lieb-Robinson bounds of power-law type for the random dimer model.
We discuss spreading estimates for dynamical systems given by the iteration of an extended CMV matrix. Using a connection due to Cantero--Grünbaum--Moral--Velázquez, this enables us to study spreading rates for quantum walks in one spatial dimension. We prove several general results which establish quantitative upper and lower bounds on the spreading of a quantum walk in terms of estimates on a pair of associated matrix cocycles. To demonstrate the power and utility of these methods, we apply them to several concrete cases of interest. In the case where the coins are distributed according to an element of the Fibonacci subshift, we are able to rather completely describe the dynamics in a particular asymptotic regime. As a pleasant consequence, this supplies the first concrete example of a quantum walk with anomalous transport, to the best of our knowldege. We also prove ballistic transport for a quantum walk whose coins are periodically distributed.
We consider Schrödinger operators on the real line with limit-periodic potentials and show that, generically, the spectrum is a Cantor set of zero Lebesgue measure and all spectral measures are purely singular continuous. Moreover, we show that for a dense set of limit-periodic potentials, the spectrum of the associated Schrödinger operator has Hausdorff dimension zero. In both results one can introduce a coupling constant $λ\in (0,\infty)$, and the respective statement then holds simultaneously for all values of the coupling constant.
We show that under natural technical conditions, the sum of a $C^2$ dynamically defined Cantor set with a compact set in most cases (for almost all parameters) has positive Lebesgue measure, provided that the sum of the Hausdorff dimensions of these sets exceeds one. As an application, we show that for many parameters, the Square Fibonacci Hamiltonian has spectrum of positive Lebesgue measure, while at the same time the density of states measure is purely singular.
This article provides a brief introductory account of the theory of aperiodic order.
We consider the iteration of a unitary operator on a separable Hilbert space and study the spreading rates of the associated discrete-time dynamical system relative to a given orthonormal basis. We prove lower bounds for the transport exponents, which measure the time-averaged spreading on a power-law scale, in terms of dimensional properties of the spectral measure associated with the unitary operator and the initial state. These results are the unitary analog of results established in recent years for the dynamics of the Schrödinger equation, which is a continuum-time dynamical system associated with a self-adjoint operator. We discuss how these general results may be studied by means of subordinacy theory in cases where the unitary operator is given by a CMV matrix. An example of particular interest in which this scenario arises is given by a time-homogeneous quantum walk on the integers. For the particular case of the time-homogeneous Fibonacci quantum walk, we illustrate how these components work together and produce explicit lower bounds for the transport exponents associated with this model.
We consider discrete Schrödinger operators with pattern Sturmian potentials. This class of potentials strictly contains the class of Sturmian potentials, for which the spectral properties of the associated Schrödinger operators are well understood. In particular, it is known that for every Sturmian potential, the associated Schrödinger operator has zero-measure spectrum and purely singular continuous spectral measures. We conjecture that the same statements hold in the more general class of pattern Sturmian potentials. We prove partial results in support of this conjecture. In particular, we confirm the conjecture for all pattern Sturmian potentials that belong to the family of Toeplitz sequences.
We generalize the approach to localization in one dimension introduced by Kunz-Souillard, and refined by Delyon-Kunz-Souillard and Simon, in the early 1980's in such a way that certain correlations are allowed. Several applications of this generalized Kunz-Souillard method to almost periodic Schrödinger operators are presented. On the one hand, we show that the Schrödinger operators on $l^2(\mathbb{Z})$ with limit-periodic potential that have pure point spectrum form a dense subset in the space of all limit-periodic Schrödinger operators on $l^2(\mathbb{Z})$. More generally, for any bounded potential, one can find an arbitrarily small limit-periodic perturbation so that the resulting operator has pure point spectrum. Our result complements the known denseness of absolutely continuous spectrum and the known genericity of singular continuous spectrum in the space of all limit-periodic Schrödinger operators on $l^2(\mathbb{Z})$. On the other hand, we show that Schrödinger operators on $l^2(\mathbb{Z})$ with arbitrarily small one-frequency quasi-periodic potential may have pure point spectrum for some phases. This was previously known only for one-frequency quasi-periodic potentials with $\|\cdot\|_\infty$ norm exceeding $2$, namely the super-critical almost Mathieu operator with a typical frequency and phase. Moreover, this phenomenon can occur for any frequency, whereas no previous quasi-periodic potential with Liouville frequency was known that may admit eigenvalues for any phase.
In this survey we discuss spectral and quantum dynamical properties of discrete one-dimensional Schrödinger operators whose potentials are obtained by real-valued sampling along the orbits of an ergodic invertible transformation. After an introductory part explaining basic spectral concepts and fundamental results, we present the general theory of such operators, and then provide an overview of known results for specific classes of potentials. Here we focus primarily on the cases of random and almost periodic potentials.
We study the Cauchy problem for the KdV equation $\partial_t u - 6 u \partial_x u + \partial_x^3 u = 0$ with almost periodic initial data $u(x,0)=V(x)$. We consider initial data $V$, for which the associated Schrödinger operator is absolutely continuous and has a spectrum that is not too thin in a sense we specify, and show the existence, uniqueness, and almost periodicity in time of solutions. This establishes a conjecture of Percy Deift for this class of initial data. The result is shown to apply to all small analytic quasiperiodic initial data with Diophantine frequency vector.
This article presents a very gentle introduction to the field of aperiodic order, aimed at a general audience. It is intended to provide a "Snapshot of Modern Mathematics" relating to the Oberwolfach mini-workshop "Dynamical versus Diffraction Spectra in the Theory of Quasicrystals" in November/December 2014.
We consider the KdV equation $$ \partial_t u +\partial^3_x u +u\partial_x u=0 $$ with quasi-periodic initial data whose Fourier coefficients decay exponentially and prove existence and uniqueness, in the class of functions which have an expansion with exponentially decaying Fourier coefficients, of a solution on a small interval of time, the length of which depends on the given data and the frequency vector involved. For a Diophantine frequency vector and for small quasi-periodic data (i.e., when the Fourier coefficients obey $|c(m)| \le \varepsilon \exp(-κ_0 |m|)$ with $\varepsilon > 0$ sufficiently small, depending on $κ_0 > 0$ and the frequency vector), we prove global existence and uniqueness of the solution. The latter result relies on our recent work \cite{DG} on the inverse spectral problem for the quasi-periodic Schrödinger equation.