On the inverse scattering problem for Jacobi matrices with the spectrum on an interval, a finite system of intervals or a Cantor set of positive length
Solving inverse scattering problem for a discrete Sturm-Liouville operator with the fast decreasing potential one gets reflection coefficients $s_\pm$ and invertible operators $I+H_{s_\pm}$, where $ H_{s_\pm}$ is the Hankel operator related to the symbol $s_\pm$. The Marchenko-Fadeev theorem (in the continuous case) and the Guseinov theorem (in the discrete case), guarantees the uniqueness of solution of the inverse scattering problem. In this article we asks the following natural question --- can one find a precise condition guaranteeing that the inverse scattering problem is uniquely solvable and that operators $I+H_{s_\pm}$ are invertible? Can one claim that uniqueness implies invertibility or vise versa?
Moreover we are interested here not only in the case of decreasing potential but also in the case of asymptotically almost periodic potentials. So we merege here two mostly developed cases of inverse problem for Sturm-Liouville operators: the inverse problem with (almost) periodic potential and the inverse problem with the fast decreasing potential.
We consider continuous cocycles arising from CMV and Jacobi matrices. Assuming the Verblunsky and Jacobi coefficients arise from generalized skew-shifts, we prove that uniform hyperbolicity of the associated cocycles is $C^0$-dense. This implies that the associated CMV and Jacobi matrices have Cantor spectrum for a generic continuous sampling map.
Let $K(γ)$ be the weakly equilibrium Cantor type set introduced in [10]. It is proven that the monic orthogonal polynomials $Q_{2^s}$ with respect to the equilibrium measure of $K(γ)$ coincide with the Chebyshev polynomials of the set. Procedures are suggested to find $Q_{n}$ of all degrees and the corresponding Jacobi parameters. It is shown that the sequence of the Widom factors is bounded below.
We establish Lieb-Thirring power bounds on discrete eigenvalues of Jacobi operators for Schatten class perturbations under very general assumptions. Our results apply, in particular, to perturbations of reflectionless Jacobi operators with finite gap and Cantor-type essential spectrum.
We study one-dimensional random Jacobi operators corresponding to strictly ergodic dynamical systems. In this context, we characterize the spectrum of these operators by non-uniformity of the transfer matrices and the set where the Lyapunov exponent vanishes. Adapting this result to subshifts satisfying the so-called Boshernitzan condition, it turns out that the spectrum is supported on a Cantor set with Lebesgue measure zero. This generalizes earlier results for Schrödinger operators.
We study the properties and asymptotics of the Jacobi matrices associated with equilibrium measures of the weakly equilibrium Cantor sets. These family of Cantor sets were defined and different aspects of orthogonal polynomials on them were studied recently. Our main aim is numerically examine some conjectures concerning orthogonal polynomials which do not directly follow from previous results. We also compare our results with more general conjectures made for recurrence coefficients associated with fractal measures supported on $\mathbb{R}$.
We consider infinite quasi-periodic Jacobi self-adjoint matrices for which the three main diagonals are given via values of real analytic functions on the trajectory of the shift $x\rightarrow x+ω$. We assume that the Lyapunov exponent $L(E_{0})$ of the corresponding Jacobi cocycle satisfies $L(E_{0})\geγ>0$. In this setting we prove that the number of eigenvalues $E_{j}^{(n)}(x)$ of a submatrix of size $n$ contained in an interval $I$ centered at $E_{0}$ with $|I|=n^{-C_{1}}$ does not exceed $(\log n)^{C_{0}}$ for any $x$. Here $n\ge n_{0}$, and $n_{0}$, $C_{0}$, $C_{1}$ are constants dependi…
This paper investigates the spectral properties of Jacobi matrices with limit-periodic coefficients. We show that for a residual set of such matrices, the spectrum is a Cantor set of zero Lebesgue measure, and the spectral measures are purely singular continuous. For a dense set of limit-periodic Jacobi matrices we can strengthen the result and show that the spectrum is a Cantor set of zero lower box counting dimension, and hence in particular of zero Hausdorff dimension, while still retaining the singular continuity of the spectral type. We also show how results of this nature can be establis…
We present a comprehensive treatment of relative oscillation theory for finite Jacobi matrices. We show that the difference of the number of eigenvalues of two Jacobi matrices in an interval equals the number of weighted sign-changes of the Wronskian of suitable solutions of the two underlying difference equations. Until now only the case of perturbations of the main diagonal was known. We extend the known results to arbitrary perturbations, allow any (half-)open and closed spectral intervals, simplify the proof, and establish the comparison theorem.
In this paper we introduce and study a certain intricate Cantor-like set $C$ contained in unit interval. Our main result is to show that the set $C$ itself, as well as the set of dissipative points within $C$, both have Hausdorff dimension equal to 1. The proof uses the transience of a certain non-symmetric Cauchy-type random walk.
We study the limit behavior of the Dirichlet and Neumann eigenvalue counting function of generalized second order differential operators $\frac{d}{d μ} \frac{d}{d x}$, where $μ$ is a finite atomless Borel measure on some compact interval $[a,b]$. We firstly recall the results of the spectral asymptotics for these operators received so far. Afterwards, we give the spectral asymptotics for so called random recursive Cantor measures. Finally, we compare the results for random recursive and random homogeneous Cantor measures.
For the square tridiagonal Fibonacci Hamiltonian, we prove existence of an open set of parameters which yield mixed interval-Cantor spectra (i.e. spectra containing an interval as well as a Cantor set), as well as mixed density of states measure (i.e. one whose absolutely continuous and singular continuous components are both nonzero). Using the methods developed in this paper, we also show existence of parameter regimes for the square continuum Fibonacci Schrodinger operator yielding mixed interval-Cantor spectra. These examples provide the first explicit examples of an interesting phenomenon…
We study the limiting behavior of the Dirichlet and Neumann eigenvalue counting function of generalized second order differential operators $\frac{d}{d μ} \frac{d}{d x}$, where $μ$ is a finite atomless Borel measure on some compact interval $[a,b]$. Therefore, we firstly recall the results of the spectral asymptotics for these operators received so far. Afterwards, we make a proposition about the convergence behavior for so called random $V$-variable Cantor measures.
We consider the 1D periodic Jacobi matrices. The spectrum of this operator is purely absolutely continuous and consists of intervals separated by gaps. We solve the inverse problem (including characterization) in terms of vertical slits on the quasimomentum domain . Furthermore, we obtain a priori two-sided estimates for vertical slits in terms of Jacoby matrices.
One of the first theorems in perturbation theory claims that for an arbitrary self-adjoint operator A there exists a perturbation B of Hilbert-Schmidt class, which destroys completely the absolutely continuous spectrum of A (von Neumann). However, if A is the discrete free 1-D Schrödinger operator and B is a Jacobi matrix the a.c. spectrum remains perfectly the same. Moreover, Killip and Simon described explicitly the spectral properties for such A+B. Jointly with Damanik they generalized this result to the case of perturbations of periodic Jacobi matrices. Recall that the spectrum of a period…
Jacobi matrices probably are the most classical object in spectral theory, while CMV matrices are a comparably fresh one, although they are related to a very classical topic, namely to orhtogonal polynomials on the unit circle (in the same way as Jacobi matrices are related to orhogonal polynomials on the real axis). We will discuss the third member of this family. Our matrices are generated by the orthonormal systems of functions related to the so-called Strong Moment Problem. For this reason we call them SMP matrices. For instance, one can describe the spectral sets of periodic SMP matrices.…
One of the first and therefore most important theorems in perturbation theory claims that for an arbitrary self-adjoint operator A there exists a perturbation B of Hilbert-Schmidt class with arbitrary small operator norm, which destroys completely the absolutely continuos (a.c.) spectrum of the initial operator A (von Neumann). However, if A is the discrete free 1-D Schrödinger operator and B is an arbitrary Jacobi matrix (of Hilbert-Schmidt class) the a.c. spectrum remains perfectly the same, that is, the interval [-2,2]. Moreover, Killip and Simon described explicitly the spectral properties…
We consider Jacobi matrices and Schrodinger operators that are reflectionless on an interval. We give a systematic development of a certain parametrization of this class, in terms of suitable spectral data, that is due to Marchenko. Then some applications of these ideas are discussed.
We consider Jacobi matrices whose essential spectrum is a finite union of closed intervals. We focus on Szego's theorem, Jost solutions, and Szego asymptotics for this situation. This announcement describes talks the authors gave at OPSFA 2007.
This is the second part of the paper arXiv:1309.0959v2 on the theory of SMP (Strong Moment Problem) matrices and their relation to the Killip-Simon problem on two disjoint intervals. In this part we define and study the Jacobi flow on SMP matrices.
We present certain results on the direct and inverse spectral theory of the Jacobi operator with complex periodic coefficients. For instance, we show that any $N$-th degree polynomial whose leading coefficient is $(-1)^N$ is the Hill discriminant of finitely many discrete $N$-periodic Schrödinger operators (Theorem 1). Also, in the case where the spectrum is a closed interval we prove a result (Theorem 5) which is the analog of Borg's Theorem for the non-self-adjoint Jacobi case.
We consider self-adjoint unbounded Jacobi matrices with diagonal q_n=n and weights λ_n=c_n n, where c_n is a 2-periodical sequence of real numbers. The parameter space is decomposed into several separate regions, where the spectrum is either purely absolutely continuous or discrete. This constitutes an example of the spectral phase transition of the first order. We study the lines where the spectral phase transition occurs, obtaining the following main result: either the interval (-\infty;1/2) or the interval (1/2;+\infty) is covered by the absolutely continuous spectrum, the remainder of the …
We consider Jacobi matrices with zero diagonal and off-diagonals given by elements of the hull of the Fibonacci sequence and show that the spectrum has zero Lebesgue measure and all spectral measures are purely singular continuous. In addition, if the two hopping parameters are distinct but sufficiently close to each other, we show that the spectrum is a dynamically defined Cantor set, which has a variety of consequences for its local and global fractal dimension.
We continue studying the connection between Jacobi matrices defined on a tree and multiple orthogonal polynomials (MOPs) that was discovered previously by the authors. In this paper, we consider Angelesco systems formed by two analytic weights and obtain asymptotics of the recurrence coefficients and strong asymptotics of MOPs along all directions (including the marginal ones). These results are then applied to show that the essential spectrum of the related Jacobi matrix is the union of intervals of orthogonality.
