Alexei Poltoratski, Christian Remling
We study spaces of reflectionless Jacobi matrices. The main theme is the following type of question: Given a reflectionless Jacobi matrix, is it possible to approximate it by other reflectionless and, typically, simpler Jacobi matrices of a special type? For example, can we approximate by periodic operators?
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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 prove that Jacobi, CMV, and Schrödinger operators, which are reflectionless on a homogeneous set E (in the sense of Carleson), under the assumption of a Blaschke-type condition on their discrete spectra accumulating at E, have purely absolutely continuous spectrum on E.
If a Jacobi matrix $J$ is reflectionless on $(-2,2)$ and has a single $a_{n_0}$ equal to 1, then $J$ is the free Jacobi matrix $a_n\equiv 1$, $b_n\equiv 0$. I'll discuss this result and its generalization to arbitrary sets and present several applications, including the following: if a Jacobi matrix has some portion of its $a_n$'s close to 1, then one assumption in the Denisov-Rakhmanov Theorem can be dropped.
We provide a concise, yet fairly complete discussion of the concept of essential closures of subsets of the real axis and their intimate connection with the topological support of absolutely continuous measures. As an elementary application of the notion of the essential closure of subsets of $\bbR$ we revisit the fact that CMV, Jacobi, and Schrödinger operators, reflectionless on a set E of positive Lebesgue measure, have absolutely continuous spectrum on the essential closure of the set E (with uniform multiplicity two on E). Though this result in the case of Schrödinger and Jacobi operators…
The property that a Jacobi matrix is reflectionless is usually characterized either in terms of Weyl m-functions or the vanishing of the real part of the boundary values of the diagonal matrix elements of the resolvent. We introduce a characterization in terms of stationary scattering theory (the vanishing of the reflection coefficients) and prove that this characterization is equivalent to the usual ones. We also show that the new characterization is equivalent to the notion of being dynamically reflectionless, thus providing a short proof of an important result of [Breuer-Ryckman-Simon]. The…
We study several related aspects of reflectionless Jacobi matrices. Our first set of results deals with the singular part of reflectionless measures. We then introduce and discuss Lyapunov exponents, density of states measures, and other related quantities in a \textit{general} setting. This is related to the previous material because the density of states measures are reflectionless on certain sets.
Reflectionless CMV matrices are studied using scattering theory. By changing a single Verblunsky coefficient a full-line CMV matrix can be decoupled and written as the sum of two half-line operators. Explicit formulas for the scattering matrix associated to the coupled and decoupled operators are derived. In particular, it is shown that a CMV matrix is reflectionless iff the scattering matrix is off-diagonal which in turn provides a short proof of an important result of [Breuer-Ryckman-Simon]. These developments parallel those recently obtained for Jacobi matrices.
We study the entropic fluctuations of a general XY spin chain where initially the left(x<0)/right(x>0) part of the chain is in thermal equilibrium at inverse temperature Tl/Tr. The temperature differential results in a non-trivial energy/entropy flux across the chain. The Evans-Searles (ES) entropic functional describes fluctuations of the flux observable with respect to the initial state while the Gallavotti-Cohen (GC) functional describes these fluctuations with respect to the steady state (NESS) the chain reaches in the large time limit. We also consider the full counting statistics (FCS) o…
We study CMV matrices by focusing on their right-limit sets. We prove a CMV version of a recent result of Remling dealing with the implications of the existence of absolutely continuous spectrum, and we study some of its consequences. We further demonstrate the usefulness of right limits in the study of weak asymptotic convergence of spectral measures and ratio asymptotics for orthogonal polynomials by extending and refining earlier results of Khrushchev. To demonstrate the analogy with the Jacobi case, we recover corresponding previous results of Simon using the same approach.
A note on the A.C. spectrum of limit-periodic Jacobi matrices
该研究针对弱指数类EC(η)的极限周期雅可比矩阵，通过建立谱齐次性、传递无反射性并应用Poltoratski与Remling定理，证明η足够大时其具有纯绝对连续谱型。
Clock spacing for two-sided Jacobi matrices
研究具有两个可移动端点的有限截断双边雅可比矩阵的局部特征值间距，证明点态无反射条件下时钟间距的类似性质，并得到含可移动起点的Christoffel-Darboux核的新标度极限。
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.
Jacobi matrices with lacunary spectrum
该研究在附加增长条件下推导缺项谱Jacobi矩阵的元素渐近性并证明逆结果，同时探究其与典范系统、de Branges空间的关联，拓展了谱理论相关研究。
The Kotani-Last conjecture states that every ergodic operator in one space dimension with non-empty absolutely continuous spectrum must have almost periodic coefficients. This statement makes sense in a variety of settings; for example, discrete Schrödinger operators, Jacobi matrices, CMV matrices, and continuum Schrödinger operators. In the main body of this paper we show how to construct counterexamples to the Kotani-Last conjecture for continuum Schrödinger operators by adapting the approach developed by Volberg and Yuditskii to construct counterexamples to the Kotani-Last conjecture for Ja…
Borg-type uniqueness theorems for matrix-valued Jacobi operators H and supersymmetric Dirac difference operators D are proved. More precisely, assuming reflectionless matrix coefficients A, B in the self-adjoint Jacobi operator H=AS^+ + A^-S^- + B (with S^\pm the right/left shift operators on the lattice Z) and the spectrum of H to be a compact interval [E_-,E_+], E_- < E_+, we prove that A and B are certain multiples of the identity matrix. An analogous result which, however, displays a certain novel nonuniqueness feature, is proved for supersymmetric self-adjoint Dirac difference operators D…
We study spectrum inclusion regions for complex Jacobi matrices which are compact perturbations of real periodic Jacobi matrices. The condition sufficient for the lack of discrete spectrum for such matrices is given
The paper contains two results on the equivalence classes of block Jacobi matrices: first, that the Jacobi matrix of type 2 in the Nevai class has A_n coefficients converging to 1, and second, that under an L1-type condition on the Jacobi coefficients, equivalent Jacobi matrices of type 1, 2 and 3 are pairwise asymptotic.
In this paper, we study complex Jacobi matrices obtained by the Christoffel and Geronimus transformations at a nonreal complex number, including the properties of the corresponding sequences of orthogonal polynomials. We also present some invariant and semi-invariant properties of Jacobi matrices under such transformations. For instance, we show that a Nevai class is invariant under the transformations in question, which is not true in general, and that the ratio asymptotic still holds outside the spectrum of the corresponding symmetric complex Jacobi matrix but the spectrum could include one …
For full-line Jacobi matrices, Schrödinger operators, and CMV matrices, we show that being reflectionless, in the sense of the well-known property of $m$-functions, is equivalent to a lack of reflection in the dynamics in the sense that any state that goes entirely to $x=-\infty$ as $t\to -\infty $ goes entirely to $x=\infty$ as $t\to\infty$. This allows us to settle a conjecture of Deift and Simon from 1983 regarding ergodic Jacobi matrices.
We study spectrum inclusion regions for complex Jacobi matrices which are compact perturbations of the discrete laplacian. The condition sufficient for the lack of discrete spectrum for such matrices is given.
We study Jacobi matrices on trees whose coefficients are generated by multiple orthogonal polynomials. Hilbert space decomposition into an orthogonal sum of cyclic subspaces is obtained. For each subspace, we find generators and the generalized eigenfunctions written in terms of the orthogonal polynomials. The spectrum and its spectral type are studied for large classes of orthogonality measures.
We compute the asymptotics of eigenvalues of Jacobi matrices with the zero coefficients on the main diagonal and the off-diagonal coefficients which converge to zero.
We prove that on the spectrum the integrated density of states (IDS for short) of periodic Jacobi matrices is related to the discriminant. The method is to count the number of generalized zeros of Bloch wave solutions.
Ambarzumian-type mixed inverse spectral problems for Jacobi matrices
本文研究Jacobi矩阵的Ambarzumian型混合逆谱问题，探讨在已知除前m个对角元素外的所有元素及m个有序特征值时矩阵的唯一确定性。
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 develop relative oscillation theory for Jacobi matrices which, rather than counting the number of eigenvalues of one single matrix, counts the difference between the number of eigenvalues of two different matrices. This is done by replacing nodes of solutions associated with one matrix by weighted nodes of Wronskians of solutions of two different matrices.
We look at periodic Jacobi matrices on trees. We provide upper and lower bounds on the gap of such operators analogous to the well known gap in the spectrum of the Laplacian on the upper half-plane with hyperbolic metric. We make some conjectures about antibound states and make an interesting observation for what [3] calls the rg-model.
In this work, we explicitly compute the group inverse of symmetric and periodic Jacobi matrices.
We study stability of spectral types for semi-infinite self-adjoint tridiagonal matrices under random decaying perturbations. We show that absolutely continuous spectrum associated with bounded eigenfunctions is stable under Hilbert-Schmidt random perturbations. We also obtain some results for singular spectral types.
We study spectral properties of unbounded Jacobi matrices with periodically modulated or blended entries. Our approach is based on uniform asymptotic analysis of generalized eigenvectors. We determine when the studied operators are self-adjoint. We identify regions where the point spectrum has no accumulation points. This allows us to completely describe the essential spectrum of these operators.
A complete characterization is provided of Hankel matrices commuting with Jacobi matrices which correspond to hypergeometric orthogonal polynomials from the Askey scheme. It follows, as the main result of the paper, that the generalized Hilbert matrix is the only prominent infinite-rank Hankel matrix which, if regarded as an operator on $\ell^{2}(\mathbb{N}_{0})$, is diagonalizable by application of the commutator method with Jacobi matrices from the mentioned families.
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.…
A result of Borg--Hochstadt in the theory of periodic Jacobi matrices states that such a matrix has constant diagonals as long as all gaps in its spectrum are closed (have zero length). We suggest a quantitative version of this result by proving the two-sided bounds between oscillations of the matrix entries along the diagonals and the length of the maximal gap in the spectrum.
We study spectral properties of bounded and unbounded complex Jacobi matrices. In particular, we formulate conditions assuring that the spectrum of the studied operators is continuous on some subsets of the complex plane and we provide uniform asymptotics of their generalised eigenvectors. We illustrate our results by considering complex perturbations of real Jacobi matrices belonging to several classes: asymptotically periodic, periodically modulated and the blend of these two. Moreover, we provide conditions implying existence of a unique closed extension. The method of the proof is based on…
The discrete spectrum of complex Jacobi matrices that are compact perturbations of the discrete laplacian is under consideration. The rate of stabilization for the the matrix entries which provides finiteness of the discrete spectrum and is sharp in the sense of order is found. The Jacobi matrix with the discrete spectrum having a single limit point is constructed. The results can be viewed as the discrete analogs of the well known theorems by B.S. Pavlov about Schrödinger operators on the half line with a complex potential.
A problem of determining zeroes of the Gauss hypergeometric function goes back to Klein, Hurwitz, and Van Vleck. In this very short note we show how ratios of hypergeometric functions arise as m-functions of Jacobi matrices and we then revisit the problem based on the recent developments of the spectral theory of non-Hermitian Jacobi matrices.
We study asymptotics of generalized eigenvectors associated with Jacobi matrices. Under weak conditions on the coefficients we identify when the matrices are self-adjoint and show that they satisfy strong non-subordinacy condition.
We introduce a class of Jacobi operators with discrete spectra which is characterized by a simple convergence condition. With any operator J from this class we associate a characteristic function as an analytic function on a suitable domain, and show that its zero set actually coincides with the set of eigenvalues of J in that domain. Further we derive sufficient conditions under which the spectrum of J is approximated by spectra of truncated finite-dimensional Jacobi matrices. As an application we construct several examples of Jacobi matrices for which the characteristic function can be expre…
We recall criteria on the spectrum of Jacobi matrices such that the corresponding isospectral torus consists of periodic operators. Motivated by those known results for Jacobi matrices, we define a new class of operators called GMP matrices. They form a certain Generalization of matrices related to the strong Moment Problem. This class allows us to give a parametrization of almost periodic finite gap Jacobi matrices by periodic GMP matrices. Moreover, due to their structural similarity we can carry over numerous results from the direct and inverse spectral theory of periodic Jacobi matrices to…
We consider the inverse dynamical problem for the dynamical system with discrete time associated with the semi-infinite Jacobi matrix. We solve the inverse problem for such a system and answer a question on the characterization of the inverse data. As a by-product we give a necessary and sufficient condition for the measure on the real line line to be the spectral measure of semi-infinite discrete Schrodinger operator.
A discrete analogue of a Schrodinger type operator proposed by J. Bellissard has a singular continuous spectrum. In this remark we answer the conjecture formulated by D. Bessis, M. Mehta and P. Moussa on the coefficients of that operator. It turns out that the coefficients have a more complicated behavior than it was conjectured.
We show that the Hausdorff dimension of the spectral measure of a class of deterministic, i. e. nonrandom, block-Jacobi matrices may be determined exactly, improving a result of Zlatos (J. Funct. Anal. 207, 216-252 (2004)).
A Weyl Matrix Perspective on Unbounded Non-Self-Adjoint Jacobi Matrices
针对Pushnitski--\u0160Stampach的编码方法，提出基于外尔函数的新视角，简化证明并推广至无界非自伴雅可比矩阵情形，建立正非对角元真雅可比矩阵与谱数据的双射且证明其连续性，还给出具独立意义的局部Borg--Marchenko定理。
We introduce a class of doubly infinite complex Jacobi matrices determined by a simple convergence condition imposed on the diagonal and off-diagonal sequences. For each Jacobi matrix belonging to this class, an analytic function, called a characteristic function, is associated with it. It is shown that the point spectrum of the corresponding Jacobi operator restricted to a suitable domain coincides with the zero set of the characteristic function. Also, coincidence regarding the order of a zero of the characteristic function and the algebraic multiplicity of the corresponding eigenvalue is pr…
We establish Lieb-Thirring power bounds on discrete eigenvalues of Jacobi operators for Schatten class complex perturbations of periodic and more generally finite gap Jacobi matrices.
Barrier nonsubordinacy and absolutely continuous spectrum of block Jacobi matrices
研究阻塞雅可比矩阵中绝对连续谱与非从属广义特征向量不存在性的关系，引入屏障非从属性概念并证明其与绝对连续性的关系，扩展一维条件至多维情况并应用于具体阻塞雅可比矩阵类。
For ergodic 1d Jacobi operators we prove that the random singular components of any spectral measure are almost surely mutually disjoint as long as one restricts to the set of positive Lyapunov exponent. In the context of extended Harper's equation this yields the first rigorous proof of the Thouless' formula for the Lyapunov exponent in the dual regions.
We provide a complete spectral analysis of all self-adjoint operators acting on $\ell^{2}(\mathbb{Z})$ which are associated with two doubly infinite Jacobi matrices with entries given by $$ q^{-n+1}δ_{m,n-1}+q^{-n}δ_{m,n+1} $$ and $$ δ_{m,n-1}+αq^{-n}δ_{m,n}+δ_{m,n+1}, $$ respectively, where $q\in(0,1)$ and $α\in\mathbb{R}$. As an application, we derive orthogonality relations for the Ramanujan entire function and the third Jackson $q$-Bessel function.
Cut a Jacobi matrix into two pieces by removing the n-th column and n-th row. We give neccessary and sufficient conditions for the spectra of the original matrix plus the spectra of the two submatrices to uniqely determine the original matrix. Our result contains Hostadt's original result as a special case.
In this paper it is considered a spectral density for a class of Jacobi matrices with absolutely continuous spectrum that was examined first by Moszynski. It is shown that the corresponding spectral density is equivalent to the positive continuous function everywhere except maybe the point $x=0$.
The paper concerns with infinite symmetric block Jacobi matrices $\bf J$ with $p\times p$-matrix entries. We present new conditions for general block Jacobi matrices to be selfadjoint and have discrete spectrum. In our previous papers there was established a close relation between a class of such matrices and symmetric $2p\times 2p$ Dirac operators $\mathrm{\bf D}_{X,α}$ with point interactions in $L^2(\Bbb R; \Bbb C^{2p})$. In particular, their deficiency indices are related by $n_\pm(\mathrm{\bf D}_{X,α})= n_\pm({\bf J}_{X,α})$. For block Jacobi matrices of this class we present several cond…
Spectral Gaps for Jacobi Matrices on Graphs
本文研究了树上周期 Jacobi 矩阵的有限商的特征值隙的界，证明了谱隙的 Alon-Boppana 型界及其他特征值隙的比较结果。
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.
Our main result asserts that a certain natural non-linear operator on Jacobi matrices built by a hyperbolic polynomial with real Julia set is a contraction in operator norm if the polynomial is sufficiently hyperbolic. This allows us to get for such polynomials the solution of a problem of Bellissard, in other words, to prove the limit periodicity of the limit Jacobi matrix. This fact does not require the iteration of the same fixed polynomial, and therefore it gives a wide class of limit periodic Jacobi matrices with singular continuous spectrum.
In this work we build a certain machine that allows to construct almost periodic Jacobi matrices with singularly continuous spectrum a prescribed p-adic hull.
Some inverse problems for semi-infinite periodic generalized Jacobi matrices are considered. In particular, a generalization of the Abel criterion is presented. The approach is based on the fact that the solvability of the Pell-Abel equation is equivalent to the existence of a certainly normalized $J$-unitary $2\times 2$-matrix polynomial (the monodromy matrix).
We consider a class of Jacobi matrices with periodically modulated diagonal in a critical hyperbolic ("double root") situation. For the model with "non-smooth" matrix entries we obtain the asymptotics of generalized eigenvectors and analyze the spectrum. In addition, we reformulate a very helpful theorem from a paper of Janas and Moszynski in its full generality in order to serve the needs of our method.
We obtain various versions of classical Lieb--Thirring bounds for one- and multi-dimensional complex Jacobi matrices. Our method is based on Fan-Mirski Lemma and seems to be fairly general.
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.
Symmetric Jacobi matrices on one sided homogeneous trees are studied. Essential selfadjointness of these matrices turns out to depend on the structure of the tree. If a tree has one end and infinitely many origin points the matrix is always essentially selfadjoint independently of the growth of its coefficients. In case a tree has one origin and infinitely many ends, the essential selfadjointness is equivalent to that of an ordinary Jacobi matrix obtained by the restriction to the so called radial functions. For nonselfadjoint matrices the defect spaces are described in terms of the Poisson ke…
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