On the location of the discrete spectrum for complex Jacobi matrices

We study spectrum inclusion regions for complex Jacobi matrices that are compact perturbations of the discrete Laplacian. The condition sufficient for the lack of a discrete spectrum for such matrices is given.

     

Unbounded Jacobi Matrices with a Few Gaps in the Essential Spectrum: Constructive Examples

We give explicit examples of unbounded Jacobi operators with a few gaps in their essential spectrum. More precisely a class of Jacobi matrices whose absolutely continuous spectrum fills any finite number of bounded intervals is considered. Their point spectrum accumulates to +?? and ???. The asymptotics of large eigenvalues is also found.     

Discrete spectrum in a critical coupling case of Jacobi matrices with spectral phase transitions by uniform asymptotic analysis

For a two-parameter family of Jacobi matrices exhibiting first-order spectral phase transitions, we prove discreteness of the spectrum in the positive real axis when the parameters are in one of the transition boundaries. To this end, we develop a method for obtaining uniform asymptotics, with respect to the spectral parameter, of the generalized eigenvectors. Our technique can be applied to a wide range of Jacobi matrices.     

Mourre's theory for some unbounded Jacobi matrices

We show a Mourre estimate for a class of unbounded Jacobi matrices. In particular, we deduce the absolute continuity of the spectrum of such matrices. We further conclude some propagation theorems for them.     

Two-weight Hilbert transform and Lipschitz property of Jacobi matrices associated to hyperbolic polynomials

We are going to prove a Lipschitz property of Jacobi matrices built by orthogonalizing polynomials with respect to measures in the orbit of classical Perron-Frobenius-Ruelle operators associated to hyperbolic polynomial dynamics. This Lipschitz estimate will not depend on the dimension of the Jacobi matrix. It is obtained using some sufficient conditions for two-weight boundedness of the Hilbert transform. It has been proved in [F. Peherstorfer, A. Volberg, P. Yuditskii, Limit periodic Jacobi matrices with prescribed p-adic hull and a singular continuous spectrum, Math. Res. Lett. 13 (2-3) (2006) 215-230] for all polynomials with sufficiently big hyperbolicity and in the most symmetric case t=0 that the Lipschitz estimate becomes exponentially better when the dimension of the Jacobi matrix grows. 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. We suggest a scheme how to approach Bellissard's problem for all hyperbolic dynamics by uniting the methods of the present paper and those of [F. Peherstorfer, A. Volberg, P. Yuditskii, Limit periodic Jacobi matrices with prescribed p-adic hull and a singular continuous spectrum, Math. Res. Lett. 13 (2-3) (2006) 215-230]. On the other hand, the nearness of Jacobi matrices under consideration in operator norm implies a certain nearness of their canonical spectral measures. One can notice that this last claim just gives us the classical commutative Perron-Frobenius-Ruelle theorem (it is concerned exactly with the nearness of such measures). In particular, in many situations we can see that the classical Perron-Frobenius-Ruelle theorem is a corollary of a certain non-commutative observation concerning the quantitative nearness of pertinent Jacobi matrices in operator norm.     

On the point spectrum of periodic Jacobi matrices with matrix entries: elementary approach

This work consists of two parts. The first one contains a characterization (localization) of the point spectrum of one sided, infinite and periodic Jacobi matrices with scalar entries. The second one deals with the same questions about one sided, infinite periodic Jacobi matrices with matrix entries. In particular, an example illustrating the difference between the above localization property in scalar and matrix entries cases is given.     

Decay Bounds on Eigenfunctions and the Singular Spectrum of Unbounded Jacobi Matrices

Bounds on the exponential decay of generalized eigenfunctionsof bounded and unbounded selfadjoint Jacobi matrices in are established. Two cases are considered separatelyand lead to different results: (i) the case in which the spectralparameter lies in a general gap of the spectrum of the Jacobimatrix and (ii) the case of a lower semibounded Jacobi matrixwith values of the spectral parameter below the spectrum. Itis demonstrated by examples that both results are sharp. Weapply these results to obtain a "many barriers-type" criterionfor the existence of square-summable generalized eigenfunctionsof an unbounded Jacobi matrix at almost every value of the spectralparameter in suitable open sets. In particular, this leads toexamples of unbounded Jacobi matrices with a spectral mobilityedge, i.e., a transition from purely absolutely continuous spectrumto dense pure point spectrum.     

Thin spectra and singular continuous spectral measures for limit-periodic Jacobi matrices

This paper investigates the spectral properties of Jacobi matrices with limit-periodic coefficients. We show that generically 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 show that the spectrum is a Cantor set of zero lower box counting dimension while still retaining the singular continuity of the spectral type. We also show how results of this nature can be established by fixing the off-diagonal coefficients and varying only the diagonal coefficients, and, in a more restricted version, by fixing the diagonal coefficients to be zero and varying only the off-diagonal coefficients. We apply these results to produce examples of weighted Laplacians on the multidimensional integer lattice having purely singular continuous spectral type and zero-dimensional spectrum.     

Non-self-adjoint Jacobi matrices with a rank-one imaginary part

We develop direct and inverse spectral analysis for finite and semi-infinite non-self-adjoint Jacobi matrices with a rank-one imaginary part. It is shown that given a set of n not necessarily distinct nonreal numbers in the open upper (lower) half-plane uniquely determines an n×n Jacobi matrix with a rank-one imaginary part having those numbers as its eigenvalues counting algebraic multiplicity. Algorithms of reconstruction for such finite Jacobi matrices are presented. A new model complementing the well-known Livsic triangular model for bounded linear operators with a rank-one imaginary part is obtained. It turns out that the model operator is a non-self-adjoint Jacobi matrix. We show that any bounded, prime, non-self-adjoint linear operator with a rank-one imaginary part acting on some finite-dimensional (respectively separable infinite-dimensional Hilbert space) is unitarily equivalent to a finite (respectively semi-infinite) non-self-adjoint Jacobi matrix. This obtained theorem strengthens a classical result of Stone established for self-adjoint operators with simple spectrum. We establish the non-self-adjoint analogs of the Hochstadt and Gesztesy-Simon uniqueness theorems for finite Jacobi matrices with nonreal eigenvalues as well as an extension and refinement of these theorems for finite non-self-adjoint tri-diagonal matrices to the case of mixed eigenvalues, real and nonreal. A unique Jacobi matrix, unitarily equivalent to the operator of integration in the Hilbert space L₂[0,l] is found as well as spectral properties of its perturbations and connections with the well-known Bernoulli numbers. We also give the analytic characterization of the Weyl functions of dissipative Jacobi matrices with a rank-one imaginary part.     

Anisotropic Jacobi matrices with absolutely continuous spectrum

We describe a class of anisotropic Jacobi matrices with absolutely continuous spectrum and discrete singular spectrum. The discrete Schrödinger operator perturbed by a potential having different behaviors near +∞ and -∞, belongs to this class. We use the method of the conjugate operator and techniques similar to those for the 3-body problem.     

Discrete spectrum in the gaps for perturbations of periodic Jacobi matrices

In the paper we study the problem of the finiteness of the discrete spectrum for operators generated by perturbations of periodic Jacobi matrices. In particular, estimate formulae for the number of the eigenvalues created by perturbations in the gaps of the unperturbed operator are established.     

Jacobi matrices with rapidly growing weights having only discrete spectrum

We establish sufficient conditions for self-adjointness on a class of unbounded Jacobi operators defined by matrices with main diagonal sequence of very slow growth and rapidly growing off-diagonal entries. With some additional assumptions, we also prove that these operators have only discrete spectrum.     

Finite gap Jacobi matrices: An announcement

We consider Jacobi matrices whose essential spectrum is a finite union of closed intervals. We focus on Szeg?’s theorem, Jost solutions, and Szeg? asymptotics for this situation. This announcement describes talks the authors gave at OPSFA 2007.     

Spectral theory for slowly oscillating potentials I. Jacobi matrices

For Jacobi matrices on the discrete half line with slowly oscillating potentials the absolutely continuous and singular spectrum is located. The results, which can be extended to perturbed periodic potentials, show that separated regions of purely absolutely continuous resp. purely singular spectrum appear. The main tools in the proof of absolute continuity are the method of subordinacy and an abstract result on the iterated diagonalization of products of 2×2-matrices, which is applied to transfer matrices. The singular spectrum is found by using a result of Simon and Spancer.     

Unbounded Jacobi matrices at critical coupling

We consider a class of Jacobi matrices with unbounded coefficients. This class is known to exhibit a first-order phase transition in the sense that, as a parameter is varied, one has purely discrete spectrum below the transition point and purely absolutely continuous spectrum above the transition point. We determine the spectral type and solution asymptotics at the transition point.     

Spectral analysis for linear semi‐infinite mass‐spring systems

We study how the spectrum of a Jacobi operator changes when this operator is modified by a certain finite rank perturbation. The operator corresponds to an infinite mass‐spring system and the perturbation is obtained by modifying one interior mass and one spring of this system. In particular, there are detailed results of what happens in the spectral gaps and which eigenvalues do not move under the modifications considered. These results were obtained by a new tecnique of comparative spectral analysis and they generalize and include previous results for finite and infinite Jacobi matrices.     

Kotani–Last problem and Hardy spaces on surfaces of Widom type

This is a small theory of non almost periodic ergodic families of Jacobi matrices with purely (however) absolutely continuous spectrum. The reason why this effect may happen is that under our “axioms” we found an analytic condition on the resolvent set that is responsible for (exactly equivalent to) this effect.     

Spectral Density of Jacobi Matrices with Small Deviations

We present several new asymptotic trace formulas for Jacobi matrices whose coefficients satisfy a small deviation condition. Our results extend most of the existing trace formulas for Jacobi matrices.     

Spectral properties of Jacobi matrices and sum rules of special form

In this article, we relate the properties of elements of a Jacobi matrix from certain class to the properties of its spectral measure. The main tools we use are the so-called sum rules introduced by Case in [Orthogonal polynomials from the viewpoint of scattering theory, J. Math. Phys. 15 (1974) 2166-2174; Orthogonal polynomials, II. J. Math. Phys. 16 (1975) 1435-1440]. Later, the sum rules were efficiently applied by Killip-Simon [Sum rules for Jacobi matrices and their applications to spectral theory. Ann. Math. 158 (2003) 253-321] to the spectral analysis of Jacobi matrices. We use a modification of the method that permits us to work with sum rules of higher orders. As a corollary of the main theorem, we obtain a counterpart of a result of Molchanov-Novitskii-Vainberg [First KdV integrals and absolutely continuous spectrum for 1-D Schrödinger operator, Comm. Math. Phys. 216 (2001) 195-213] for a “continuous” Schrödinger operator on a half-line.     

Eigenvalues for Perturbed Periodic Jacobi Matrices by the Wigner-von Neumann Approach

The Wigner-von Neumann method, which has previously been used for perturbing continuous Schrödinger operators, is here applied to their discrete counterparts. In particular, we consider perturbations of arbitrary \({T}\)-periodic Jacobi matrices. The asymptotic behaviour of the subordinate solutions is investigated, as too are their initial components, together giving a general technique for embedding eigenvalues, \({\lambda}\), into the operator’s absolutely continuous spectrum. Introducing a new rational function, \({C(\lambda; T)}\), related to the periodic Jacobi matrices, we describe the elements of the a.c. spectrum for which this construction does not work (zeros of \({C(\lambda; T)}\)); in particular showing that there are only finitely many of them.