Search

We derive necessary and sufficient conditions for universality limits for orthogonal polynomials on the real line and related systems. One of our results is that the Christoffel-Darboux kernel has sine kernel asymptotics at a point $\xi$, with regularly varying scaling, if and only if the orthogonality measure (spectral measure) has a unique tangent measure at $\xi$ and that tangent measure is the Lebesgue measure. This includes all prior results with absolutely continuous or singular measures. Our work is not limited to bulk universality; we show that the Christoffel-Darboux kernel has a regularly varying scaling limit with a nontrivial limit kernel if and only if the orthogonality measure has a unique tangent measure at $\xi$ and that tangent measure is not a point mass. The possible limit kernels correspond to homogeneous de Branges spaces; in particular, this equivalence completely characterizes several prominent universality classes such as hard edge universality, Fisher-Hartwig singularities, and jump discontinuities in the weights. The main part of the proof is the derivation of a new homeomorphism. In order to directly apply to the Christoffel-Darboux kernel, this homeomorphism is between measures and chains of de Branges spaces, not between Weyl functions and Hamiltonians. In order to handle limits with power law weights, this homeomorphism goes beyond the more common setting of Poisson-finite measures, and allows arbitrary power bounded measures., Comment: 78 pages

In general, point spectrum of an almost periodic Jacobi matrix can depend on the element of the hull. In this paper, we study the hull of the limit-periodic Jacobi matrix corresponding to the equilibrium measure of the Julia set of the polynomial $z^2-\lambda$ with large enough $\lambda$; this is the leading model in inverse spectral theory of ergodic operators with zero measure spectrum. We prove that every element of the hull has empty point spectrum.

We study Jost solutions of Schr\"odinger operators with potentials which decay with respect to a local $H^{-1}$ Sobolev norm; in particular, we generalize to this setting the results of Christ--Kiselev for potentials between the integrable and square-integrable rates of decay, proving existence of solutions with WKB asymptotic behavior on a large set of positive energies. This applies to new classes of potentials which are not locally integrable, or have better decay properties with respect to the $H^{-1}$ norm due to rapid oscillations.

We study half-line Schr\"odinger operators with locally $H^{-1}$ potentials. In the first part, we focus on a general spectral theoretic framework for such operators, including a Last--Simon-type description of the absolutely continuous spectrum and sufficient conditions for different spectral types. In the second part, we focus on potentials which are decaying in a local $H^{-1}$ sense; we establish a spectral transition between short-range and long-range potentials and an $\ell^2$ spectral transition for sparse singular potentials. The regularization procedure used to handle distributional potentials is also well suited for controlling rapid oscillations in the potential; thus, even within the class of smooth potentials, our results apply in situations which would not classically be considered decaying or even bounded.

We prove that limit-periodic Dirac operators generically have spectra of zero Lebesgue measure and that a dense set of them have spectra of zero Hausdorff dimension. The proof combines ideas of Avila from a Schr\"odinger setting with a new commutation argument for generating open spectral gaps. This overcomes an obstacle previously observed in the literature; namely, in Schr\"odinger-type settings, translation of the spectral measure corresponds to small $L^\infty$-perturbations of the operator data, but this is not true for Dirac or CMV operators. The new argument is much more model-independent. To demonstrate this, we also apply the argument to prove generic zero-measure spectrum for CMV matrices with limit-periodic Verblunsky coefficients., Comment: 25 pages

We describe an approach to universality limits for orthogonal polynomials on the real line which is completely local and uses only the boundary behavior of the Weyl m-function at the point. We show that bulk universality of the Christoffel-Darboux kernel holds for any point where the imaginary part of the m-function has a positive finite nontangential limit. This approach is based on studying a matrix version of the Christoffel-Darboux kernel and the realization that bulk universality for this kernel at a point is equivalent to the fact that the corresponding m-function has normal limits at the same point. Our approach automatically applies to other self-adjoint systems with $2\times 2$ transfer matrices such as continuum Schr\"odinger and Dirac operators. We also obtain analogous results for orthogonal polynomials on the unit circle., Comment: 29 pages. v2 also contains results for OPUC

There is a vast theory of Chebyshev and residual polynomials and their asymptotic behavior. The former ones maximize the leading coefficient and the latter ones maximize the point evaluation with respect to an $L^\infty$ norm. We study Chebyshev and residual extremal problems for rational functions with real poles with respect to subsets of $\overline{\mathbb{R}}$. We prove root asymptotics under fairly general assumptions on the sequence of poles. Moreover, we prove Szeg\H{o}--Widom asymptotics for sets which are regular for the Dirichlet problem and obey the Parreau--Widom and DCT conditions., Comment: 33 pages

We develop a theory of regularity for Dirac operators with uniformly locally square-integrable operator data. This is motivated by Stahl--Totik regularity for orthogonal polynomials and by recent developments for continuum Schr\"odinger operators, but contains significant new phenomena. We prove that the symmetric Martin function at $\infty$ for the complement of the essential spectrum has the two-term asymptotic expansion $\Im \left( z - \frac{b}{2 z}\right) + o(\frac 1z)$ as $z \to i \infty$, which is seen as a thickness statement for the essential spectrum. The constant $b$ plays the role of a renormalized Robin constant and enters a universal inequality involving the lower average $L^2$-norm of the operator data. However, we show that regularity of Dirac operators is not precisely characterized by a single scalar equality involving $b$ and is instead characterized by a family of equalities. This work also contains a sharp Combes--Thomas estimate (root asymptotics of eigensolutions), a study of zero counting measures, and applications to ergodic and decaying operator data., Comment: 34 pages

We develop a comprehensive theory of reflectionless canonical systems with an arbitrary Dirichlet-regular Widom spectrum with the Direct Cauchy Theorem property. This generalizes, to an infinite gap setting, the constructions of finite gap quasiperiodic (algebro-geometric) solutions of stationary integrable hierarchies. Instead of theta functions on a compact Riemann surface, the construction is based on reproducing kernels of character-automorphic Hardy spaces in Widom domains with respect to Martin measure. We also construct unitary character-automorphic Fourier transforms which generalize the Paley-Wiener theorem. Finally, we find the correct notion of almost periodicity which holds for canonical system parameters in Arov gauge, and we prove generically optimal results for almost periodicity for Potapov-de Branges gauge, and Dirac operators., Comment: 57 pages

In spectral theory, $j$-monotonic families of $2\times 2$ matrix functions appear as transfer matrices of many one-dimensional operators. We present a general theory of such families, in the perspective of canonical systems in Arov gauge. This system resembles a continuum version of the Schur algorithm, and allows to restore an arbitrary Schur function along the flow of associated boundary values at infinity. In addition to results in Arov gauge, this provides a gauge-independent perspective on the Krein-de Branges formula and the reflectionless property of right limits on the absolutely continuous spectrum. This work has applications to inverse spectral problems which have better behavior with respect to a normalization at an internal point of the resolvent domain., Comment: 26 pages

There is a vast theory of the asymptotic behavior of orthogonal polynomials with respect to a measure on $\mathbb{R}$ and its applications to Jacobi matrices. That theory has an obvious affine invariance and a very special role for $\infty$. We extend aspects of this theory in the setting of rational functions with poles on $\overline{\mathbb{R}} = \mathbb{R} \cup \{\infty\}$, obtaining a formulation which allows multiple poles and proving an invariance with respect to $\overline{\mathbb{R}}$-preserving M\"obius transformations. We obtain a characterization of Stahl--Totik regularity of a GMP matrix in terms of its matrix elements; as an application, we give a proof of a conjecture of Simon -- a Ces\`aro--Nevai property of regular Jacobi matrices on finite gap sets., Comment: to appear in Transactions of the AMS

We develop a theory of regularity for continuum Schr\"odinger operators based on the Martin compactification of the complement of the essential spectrum. This theory is inspired by Stahl--Totik regularity for orthogonal polynomials, but requires a different approach, since Stahl--Totik regularity is formulated in terms of the potential theoretic Green function with a pole at $\infty$, logarithmic capacity, and the equilibrium measure for the support of the measure, notions which do not extend to the case of unbounded spectra. For any half-line Schr\"odinger operator with a bounded potential (in a locally $L^1$ sense), we prove that its essential spectrum obeys the Akhiezer--Levin condition, and moreover, that the Martin function at $\infty$ obeys the two-term asymptotic expansion $\sqrt{-z} + \frac{a}{2\sqrt{-z}} + o(\frac 1{\sqrt{-z}})$ as $z \to -\infty$. The constant $a$ in that expansion plays the role of a renormalized Robin constant suited for Schr\"odinger operators and enters a universal inequality $a \le \liminf_{x\to\infty} \frac 1x \int_0^x V(t)dt$. This leads to a notion of regularity, with connections to the root asymptotics of Dirichlet solutions and zero counting measures. We also present applications to decaying and ergodic potentials., Comment: 33 pages

We consider the Cauchy problem for the KdV hierarchy -- a family of integrable PDEs with a Lax pair representation involving one-dimensional Schr\"odinger operators -- under a local in time boundedness assumption on the solution. For reflectionless initial data, we prove that the solution stays reflectionless. For almost periodic initial data with absolutely continuous spectrum, we prove that under Craig-type conditions on the spectrum, Dirichlet data evolve according to a Lipschitz Dubrovin-type flow, so the solution is uniquely recovered by a trace formula. This applies to algebro-geometric (finite gap) solutions; more notably, we prove that it applies to small quasiperiodic initial data with analytic sampling functions and Diophantine frequency. This also gives a uniqueness result for the Cauchy problem on the line for periodic initial data, even in the absence of Craig-type conditions., Comment: 22 pages, to appear in Journal of Functional Analysis

We develop the basic theory of ergodic Schr\"odinger operators, which is well known for ergodic probability measures, in the case of a base dynamics on an infinite measure space. This includes the almost sure constancy of the spectrum and the spectral type, the definition and discussion of the density of states measure and the Lyapunov exponent, as well as a version of the Pastur--Ishii theorem. We also give some counterexamples that demonstrate that some results do not extend from the finite measure case to the infinite measure case. These examples are based on some constructions in infinite ergodic theory that may be of independent interest., Comment: 23 pages

We consider Jacobi matrices with eventually increasing sequences of diagonal and off-diagonal Jacobi parameters. We describe the asymptotic behavior of the subordinate solution at the top of the essential spectrum, and the asymptotic behavior of the spectral density at the top of the essential spectrum. In particular, allowing on both diagonal and off-diagonal Jacobi parameters perturbations of the free case of the form $- \sum_{j=1}^J c_j n^{-\tau_j} + o(n^{-\tau_1-1})$ with $0 < \tau_1 < \tau_2 < \dots < \tau_J$ and $c_1>0$, we find the asymptotic behavior of the $\log$ of spectral density to order $O(\log(2-x))$ as $x$ approaches $2$. Apart from its intrinsic interest, the above results also allow us to describe the asymptotics of the spectral density for orthogonal polynomials on the unit circle with real-valued Verblunsky coefficients of the same form., Comment: 33 pages. Same results as in previous version, some expository changes; to appear in Journal of Spectral Theory

We disprove a conjecture of Breuer-Last-Simon concerning the absolutely continuous spectrum of Jacobi matrices with coefficients that obey an $\ell^2$ bounded variation condition with step $q$. We prove existence of a.c. spectrum on a smaller set than that specified by the conjecture and prove that our result is optimal., Comment: 21 pages

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., Comment: 24 pages; new version better reflects current literature

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 Schr\"odinger 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., Comment: 26 pages

We consider Schr\"odinger 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 Schr\"odinger operator has Hausdorff dimension zero. In both results one can introduce a coupling constant $\lambda \in (0,\infty)$, and the respective statement then holds simultaneously for all values of the coupling constant., Comment: 13 pages; to appear in J. Spectral Theory

We prove that the spectrum of a limit-periodic Schr\"odinger operator is homogeneous in the sense of Carleson whenever the potential obeys the Pastur--Tkachenko condition. This implies that a dense set of limit-periodic Schr\"odinger operators have purely absolutely continuous spectrum supported on a homogeneous Cantor set. When combined with work of Gesztesy--Yuditskii, this also implies that the spectrum of a Pastur--Tkachenko potential has infinite gap length whenever the potential fails to be uniformly almost periodic., Comment: To appear in Journal of Spectral Theory

Pr\"ufer variables are a standard tool in spectral theory, developed originally for perturbations of the free Schr\"odinger operator. They were generalized by Kiselev, Remling, and Simon to perturbations of an arbitrary Schr\"odinger operator. We adapt these generalized Prufer variables to the setting of Jacobi and Szeg\H{o} recursions. We present an application to random $L^2$ perturbations of Jacobi and CMV matrices, and an application to decaying oscillatory perturbations of periodic Jacobi and CMV matrices., Comment: 22 pages, added a section on random perturbation application

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 Szeg\H{o} cocycle is not uniformly hyperbolic., Comment: 15 pages; based on the appendices of arXiv:1404.7065, which were removed in passing from arXiv:1404.7065v1 to arXiv:1404.7065v2, the latter being the version accepted for publication in IMRN

We study the quasi-periodic Schr\"odinger operator $$ -\psi"(x) + V(x) \psi(x) = E \psi(x), \qquad x \in \mathbb{R} $$ in the regime of "small" $V(x) = \sum_{m\in\mathbb{Z}^\nu}c(m)\exp (2\pi i m\omega x)$, $\omega = (\omega_1, \dots, \omega_\nu) \in \mathbb{R}^\nu$, $|c(m)| \le \varepsilon \exp(-\kappa_0|m|)$. 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 $Q (x) = \sum_{m \in \mathbb{Z}^\nu} d(m) \exp (2\pi i m\omega x)$, with the same $\omega$ and with $|d(m)| \le \sqrt{2 \varepsilon} \exp(-\frac{\kappa_0}{2} |m|)$. Our derivation relies on the study of the approximation via Hill operators with potentials $\tilde V (x) = \sum_{m \in \mathbb{Z}^\nu} c(m) \exp (2 \pi i m \tilde \omega x)$, where $\tilde \omega$ is a rational approximation of $\omega$. 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 $|\omega - \tilde \omega| \thicksim \tilde T^{-\delta}$, $0 < \delta < 1/2$ is slow, on the scale of the period $\tilde T$ of the Hill operator., Comment: 53 pages, to appear in Invent. Math

We present an abstract multiscale analysis scheme for matrix functions $(H_{\varepsilon}(m,n))_{m,n\in \mathfrak{T}}$, where $\mathfrak{T}$ is an Abelian group equipped with a distance $|\cdot|$. This is an extension of the scheme developed by Damanik and Goldstein for the special case $\mathfrak{T} = \mathbb{Z}^\nu$. Our main motivation for working out this extension comes from an application to matrix functions which are dual to certain Hill operators. These operators take the form $H_{\tilde\omega}=-\frac{d^2}{dx^2} + \varepsilon U(\tilde\omega x)$, where $U$ is a real smooth function on the torus $\mathbb{T}^\nu$, $\tilde\omega\in \mathbb{R}^\nu$ is a vector with rational components, and $\varepsilon$ is a small parameter. The group in this particular case is the quotient $\mathfrak{T} = \mathbb{Z}^\nu/\{m\in\mathbb{Z}^\nu:m\tilde\omega=0\}$. We show that the general theory indeed applies to this special case, provided that the rational frequency vector $\tilde\omega$ obeys a suitable Diophantine condition in a large box of modes. Despite the fact that in this setting the orbits $k + m\omega$, $k \in \mathbb{R}$, $m\in\mathbb{Z}^\nu$ are not dense, the dual eigenfunctions are exponentially localized and the eigenvalues of the operators can be described as $E(k+m\omega)$ with $E(k)$ being a "nice" monotonic function of the impulse $k \ge 0$. This enables us to derive a description of the Floquet solutions and the band-gap structure of the spectrum, which we will use in a companion paper to develop a complete inverse spectral theory for the Sturm-Liouville equation with small quasi-periodic potential via periodic approximation of the frequency. The analysis of the gaps in the range of the function $E(k)$ plays a crucial role in this approach., Comment: 53 pages. This paper generalizes the results from arXiv:1209.4331 to general Abelian groups

We consider the quasi-periodic Schr\"odinger operator $$ [H \psi](x) = -\psi"(x) + V(x) \psi(x) $$ in $L^2(\mathbb{R})$, where the potential is given by $$ V(x) = \sum_{m \in \mathbb{Z}^\nu \setminus \{ 0 \}} c(m)\exp (2\pi i m \omega x) $$ with a Diophantine frequency vector $\omega = (\omega_1, \dots, \omega_\nu) \in \mathbb{R}^\nu$ and exponentially decaying Fourier coefficients $|c(m)| \le \varepsilon \exp(-\kappa_0|m|)$. In the regime of small $\varepsilon > 0$ we show that the spectrum of the operator $H$ is homogeneous in the sense of Carleson., Comment: 10 pages, to appear in J. Spectr. Theory

We investigate quantum dynamics with the underlying Hamiltonian being a Jacobi or a block Jacobi matrix with the diagonal and the off-diagonal terms modulated by a periodic or a limit-periodic sequence. In particular, we investigate the transport exponents. In the periodic case we demonstrate ballistic transport, while in the limit-periodic case we discuss various phenomena such as quasi-ballistic transport and weak dynamical localization. We also present applications to some quantum many body problems. In particular, we establish for the anisotropic XY chain on $\mathbb{Z}$ with periodic parameters an explicit strictly positive lower bound for the Lieb-Robinson velocity., Comment: 26 pages, to appear in Commun. Math. Phys

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 $x-vt^\alpha$ with $0<\alpha<1$. In fact, we can characterize the values of $\alpha$ for which such a bound holds as those exceeding $\alpha_u^+$, 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., Comment: 21 pages, Final version to appear in J. Spectr. Theory

We consider products of the matrices associated with the Szeg\H{o} recursion from the theory of orthogonal polynomials on the unit circle and show that under suitable assumptions, their norms grow exponentially in the number of factors. In the language of dynamical systems, this result expresses a uniform hyperbolicity statement. We present two applications of this result. On the one hand, we identify explicitly the almost sure spectrum of extended CMV matrices with non-negative random Verblunsky coefficients. On the other hand, we show that no Ising model in one dimension exhibits a phase transition. Also, in the case of dynamically generated interaction couplings, we describe a gap labeling theorem for the Lee-Yang zeros in the thermodynamic limit., Comment: 15 pages, to appear in Int. Math. Res. Not

We prove the following higher-order Szego theorems: if a measure on the unit circle has absolutely continuous part $w(\theta)$ and Verblunsky coefficients $\alpha$ with square-summable variation, then for any positive integer $m$, $\int (1-\cos \theta)^m \log w(\theta) d\theta$ is finite if and only if $\alpha \in \ell^{2m+2}$. This is the first known equivalence result of this kind in the regime of very slow decay, i.e. with $\ell^p$ conditions with arbitrarily large $p$. The usual difficulty of controlling higher-order sum rules is avoided by a new test sequence approach., Comment: 13 pages, to appear in Constr. Approx

We consider decaying oscillatory perturbations of periodic Schr\"odinger operators on the half line. More precisely, the perturbations we study satisfy a generalized bounded variation condition at infinity and an $L^p$ decay condition. We show that the absolutely continuous spectrum is preserved, and give bounds on the Hausdorff dimension of the singular part of the resulting perturbed measure. Under additional assumptions, we instead show that the singular part embedded in the essential spectrum is contained in an explicit countable set. Finally, we demonstrate that this explicit countable set is optimal. That is, for every point in this set there is an open and dense class of periodic Schr\"odinger operators for which an appropriate perturbation will result in the spectrum having an embedded eigenvalue at that point., Comment: 17 pages

Recent results of Denisov and Kaluzhny-Shamis describe the absolutely continuous spectrum of Jacobi matrices with coefficients that obey an l^2 bounded variation condition with step p and are asymptotically periodic. We extend these results to orthogonal polynomials on the unit circle. We also replace the asymptotic periodicity condition by the weaker condition of convergence to an isospectral torus and, for p=1 and p=2, we remove even that condition., Comment: 22 pages

We disprove a conjecture of Simon for higher-order Szego theorems for orthogonal polynomials on the unit circle and propose a modified version of the conjecture., Comment: 7 pages

We discuss Schr\"odinger operators on a half-line with decaying oscillatory potentials, such as products of an almost periodic function and a decaying function. We provide sufficient conditions for preservation of absolutely continuous spectrum and give bounds on the Hausdorff dimension of the singular part of the spectral measure. We also discuss the analogs for orthogonal polynomials on the real line and unit circle., Comment: 18 pages

We consider Schr\"odinger operators with potentials satisfying a generalized bounded variation condition at infinity and an $L^p$ decay condition. This class of potentials includes slowly decaying Wigner--von Neumann type potentials $\sin(ax)/x^b$ with $b>0$. We prove absence of singular continuous spectrum and show that embedded eigenvalues in the continuous spectrum can only take values from an explicit finite set. Conversely, we construct examples where such embedded eigenvalues are present, with exact asymptotics for the corresponding eigensolutions., Comment: Corrected typos; to appear in Journal of Spectral Theory

Assuming the negative part of the potential is uniformly locally $L^1$, we prove a pointwise $L^p$ estimate on derivatives of eigenfunctions of one-dimensional Schrodinger operators. In particular, if an eigenfunction is in $L^p$, then so is its derivative, for $1\le p\le \infty$., Comment: 7 pages

We consider probability measures on the real line or unit circle with Jacobi or Verblunsky coefficients satisfying an $\ell^p$ condition and a generalized bounded variation condition. This latter condition requires that a sequence can be expressed as a sum of sequences $\beta^{(l)}$, each of which has rotated bounded variation, i.e., $\sum_{n=0}^\infty | e^{i\phi_l} \beta_{n+1}^{(l)} - \beta_n^{(l)} |$ is finite for some $\phi_l$. This includes discrete Schr\"odinger operators on a half-line or line with finite linear combinations of Wigner--von Neumann type potentials. For the real line, we prove that in the Lebesgue decomposition $d\mu=f dm + d\mu_s$ of such measures, the intersection of (-2,2) with the support of $d\mu_s$ is contained in an explicit finite set S (thus, $d\mu$ has no singular continuous part), and f is continuous and non-vanishing on $(-2,2) \setminus S$. The results for the unit circle are analogous, with (-2,2) replaced by the unit circle., Comment: 28 pages, no figures