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David Damanik

Publications and source records attributed to David Damanik.

At least 145 records · Page 8Linked to original sources
arXiv2005

Schroedinger Operators With Few Bound States

We show that whole-line Schrödinger operators with finitely many bound states have no embedded singular spectrum. In contradistinction, we show that embedded singular spectrum is possible even when the bound states approach the essential spectrum exponentially fast. We also prove the following result for one- and two-dimensional Schrödinger operators, $H$, with bounded positive ground states: Given a potential $V$, if both $H\pm V$ are bounded from below by the ground-state energy of $H$, then $V\equiv 0$.

math-ph↗
arXiv2004

Schrödinger Operators with Many Bound States

Consider the Schrödinger operators $H_{\pm}=-d^2/dx^2\pm V(x)$. We present a method for estimating the potential in terms of the negative eigenvalues of these operators. Among the applications are inverse Lieb-Thirring inequalities and several sharp results concerning the spectral properties of $H_{\pm}$.

math-ph↗
arXiv2004

Lower Transport Bounds for One-Dimensional Continuum Schrödinger Operators

We prove quantum dynamical lower bounds for one-dimensional continuum Schrödinger operators that possess critical energies for which there is slow growth of transfer matrix norms and a large class of compactly supported initial states. This general result is applied to a number of models, including the Bernoulli-Anderson model with a constant single-site potential.

math-ph↗
arXiv2004

Generic singular continuous spectrum for ergodic Schrödinger operators

We consider Schrödinger operators with ergodic potential $V_ω(n)=f(T^n(ω))$, $n \in \Z$, $ω\in Ω$, where $T:Ω\to Ω$ is a non-periodic homeomorphism. We show that for generic $f \in C(Ω)$, the spectrum has no absolutely continuous component. The proof is based on approximation by discontinuous potentials which can be treated via Kotani Theory.

math.DS↗
arXiv2004

Scaling Estimates for Solutions and Dynamical Lower Bounds on Wavepacket Spreading

We establish quantum dynamical lower bounds for discrete one-dimensional Schrödinger operators in situations where, in addition to power-law upper bounds on solutions corresponding to energies in the spectrum, one also has lower bounds following a scaling law. As a consequence, we obtain improved dynamical results for the Fibonacci Hamiltonian and related models.

math-ph↗
arXiv2004

A Condition of Boshernitzan and Uniform Convergence in the Multiplicative Ergodic Theorem

This paper is concerned with uniform convergence in the multiplicative ergodic theorem on aperiodic subshifts. If such a subshift satisfies a certain condition, originally introduced by Boshernitzan, every locally constant SL(2,R)-valued cocycle is uniform. As a consequence, the corresponding Schrödinger operators exhibit Cantor spectrum of Lebesgue measure zero. An investigation of Boshernitzan's condition then shows that these results cover all earlier results of this type and, moreover, provide various new ones. In particular, Boshernitzan's condition is shown to hold for almost all circle maps and almost all Arnoux-Rauzy subshifts.

math.DS↗
arXiv2004

Ergodic Potentials With a Discontinuous Sampling Function Are Non-Deterministic

We prove absence of absolutely continuous spectrum for discrete one-dimensional Schrödinger operators on the whole line with certain ergodic potentials, $V_ω(n) = f(T^n(ω))$, where $T$ is an ergodic transformation acting on a space $Ω$ and $f: Ω\to \R$. The key hypothesis, however, is that $f$ is discontinuous. In particular, we are able to settle a conjecture of Aubry and Jitomirskaya--Mandel'shtam regarding potentials generated by irrational rotations on the torus. The proof relies on a theorem of Kotani, which shows that non-deterministic potentials give rise to operators that have no absolutely continuous spectrum.

math-ph↗
arXiv2003

Necessary and Sufficient Conditions in the Spectral Theory of Jacobi Matrices and Schrödinger Operators

We announce three results in the theory of Jacobi matrices and Schrödinger operators. First, we give necessary and sufficient conditions for a measure to be the spectral measure of a Schrödinger operator $-\f{d^2}{dx^2} +V(x)$ on $L^2 (0,\infty)$ with $V\in L^2 (0,\infty)$ and $u(0)=0$ boundary condition. Second, we give necessary and sufficient conditions on the Jacobi parameters for the associated orthogonal polynomials to have Szegő asymptotics. Finally, we provide necessary and sufficient conditions on a measure to be the spectral measure of a Jacobi matrix with exponential decay at a given rate.

math.SP↗
arXiv2003

Half-line Schrodinger Operators With No Bound States

We consider Schödinger operators on the half-line, both discrete and continuous, and show that the absence of bound states implies the absence of embedded singular spectrum. More precisely, in the discrete case we prove that if $Δ+ V$ has no spectrum outside of the interval $[-2,2]$, then it has purely absolutely continuous spectrum. In the continuum case we show that if both $-Δ+ V$ and $-Δ- V$ have no spectrum outside $[0,\infty)$, then both operators are purely absolutely continuous. These results extend to operators with finitely many bound states.

math-ph↗
arXiv2003

Power-law bounds on transfer matrices and quantum dynamics in one dimension II

We establish quantum dynamical lower bounds for a number of discrete one-dimensional Schrödinger operators. These dynamical bounds are derived from power-law upper bounds on the norms of transfer matrices. We develop further the approach from part I and study many examples. Particular focus is put on models with finitely or at most countably many exceptional energies for which one can prove power-law bounds on transfer matrices. The models discussed in this paper include substitution models, Sturmian models, a hierarchical model, the prime model, and a class of moderately sparse potentials.

math-ph↗
arXiv2002

Localization for Discrete One Dimensional Random Word Models

We consider Schrödinger operators in $\ell^2(\Z)$ whose potentials are obtained by randomly concatenating words from an underlying set $\mathcal{W}$ according to some probability measure $ν$ on $\mathcal{W}$. Our assumptions allow us to consider models with local correlations, such as the random dimer model or, more generally, random polymer models. We prove spectral localization and, away from a finite set of exceptional energies, dynamical localization for such models. These results are obtained by employing scattering theoretic methods together with Furstenberg's theorem to verify the necessary input to perform a multiscale analysis.

math-ph↗
arXiv2002

Variational Estimates for Discrete Schrödinger Operators with Potentials of Indefinite Sign

Let $H$ be a one-dimensional discrete Schrödinger operator. We prove that if $σ_{\ess} (H)\subset [-2,2]$, then $H-H_0$ is compact and $σ_{\ess}(H)=[-2,2]$. We also prove that if $H_0 + \frac14 V^2$ has at least one bound state, then the same is true for $H_0 +V$. Further, if $H_0 + \frac14 V^2$ has infinitely many bound states, then so does $H_0 +V$. Consequences include the fact that for decaying potential $V$ with $\liminf_{|n|\to\infty} |nV(n)| > 1$, $H_0 +V$ has infinitely many bound states; the signs of $V$ are irrelevant. Higher-dimensional analogues are also discussed.

math-ph↗
arXiv2002

Bound States and the Szego Condition for Jacobi Matrices and Schrodinger Operators

For Jacobi matrices with a_n = 1+(-1)^n alpha n^{-gamma}, b_n = (-1)^n beta n^{-gamma}, we study bound states and the SzegHo condition. We provide a new proof of Nevai's result that if gamma > 1/2, the Szego condition holds, which works also if one replaces (-1)^n by cos(mu n). We show that if alpha = 0, beta not equal to 0, and gamma < 1/2, the Szego condition fails. We also show that if gamma = 1, alpha and beta are small enough (beta^2 + 8 alpha^2 < 1/24 will do), then the Jacobi matrix has finitely many bound states (for alpha = 0, beta large, it has infinitely many).

math-ph↗
arXiv2002

Arnoux-Rauzy Subshifts: Linear Recurrence, Powers, and Palindromes

We consider Arnoux-Rauzy subshifts $X$ and study various combinatorial questions: When is $X$ linearly recurrent? What is the maximal power occurring in $X$? What is the number of palindromes of a given length occurring in $X$? We present applications of our combinatorial results to the spectral theory of discrete one-dimensional Schrödinger operators with potentials given by Arnoux-Rauzy sequences.

math.CO↗
arXiv2002

Power-Law Bounds on Transfer Matrices and Quantum Dynamics in One Dimension

We present an approach to quantum dynamical lower bounds for discrete one-dimensional Schrödinger operators which is based on power-law bounds on transfer matrices. It suffices to have such bounds for a nonempty set of energies. We apply this result to various models, including the Fibonacci Hamiltonian.

math-ph↗
arXiv2002

Dynamical Upper Bounds for One-Dimensional Quasicrystals

Following the Killip-Kiselev-Last method, we prove quantum dynamical upper bounds for discrete one-dimensional Schrödinger operators with Sturmian potentials. These bounds hold for sufficiently large coupling, almost every rotation number, and every phase.

math-ph↗
arXiv2002

Log-dimensional spectral properties of one-dimensional quasicrystals

We consider discrete one-dimensional Schrödinger operators on the whole line and establish a criterion for continuity of spectral measures with respect to $\log$-Hausdorff measures. We apply this result to operators with Sturmian potentials and thereby prove logarithmic quantum dynamical lower bounds for all coupling constants and almost all rotation numbers, uniformly in the phase.

math-ph↗