Asymptotics of large eigenvalues for some discrete unbounded Jacobi matrices

The aim of this paper is to find asymptotic formulas for eigenvalues of self-adjoint discrete operators in given by some infinite symmetric Jacobi matrices. The approach used to calculate an asymptotic behaviour of eigenvalues is based on method of diagonalization, Janas and Naboko’s lemma [J. Janas, S. Naboko, Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach, SIAM J. Math. Anal. 36(2) (2004) 643–658] and the Rozenbljum theorem [G.V. Rozenbljum, Near-similarity of operators and the spectral asymptotic behaviour of pseudodifferential operators on the circle, (Russian) Trudy Maskov. Mat. Obshch. 36 (1978) 59–84]. The asymptotic formulas are given with use of eigenvalues and determinants of finite tridiagonal matrices.     

Alternative approaches to asymptotic behaviour of eigenvalues of some unbounded Jacobi matrices

In this article we calculate the asymptotic behaviour of the point spectrum for some special self-adjoint unbounded Jacobi operators J   acting in the Hilbert space l²=l²(N)$l^2=l^2(\mathbb{N})$. For given sequences of positive numbers _λn${\lambda }_n$ and real _qn$q_n$ the Jacobi operator is given by J=SW+WS^*+Q$J=\text{<italic>SW</italic>}+{\mathit{WS}}^{*}+Q$, where Q=diag(_qn)$Q=\mathrm{diag}(q_n)$ and W=diag(_λn)$W=\mathrm{diag}({\lambda }_n)$ are diagonal operators, S is the shift operator and the operator J   acts on the maximal domain. We consider a few types of the sequences {_qn}$\{q_n\}$ and {_λn}$\{{\lambda }_n\}$ and present three different approaches to the problem of the asymptotics of eigenvalues of various classes of J's. In the first approach to asymptotic behaviour of eigenvalues we use a method called successive diagonalization, the second approach is based on analytical models that can be found for some special J's and the third method is based on an abstract theorem of Rozenbljum.     

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.     

Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles

We develop a tool to approximate the entries of a large dimensional complex Jacobi ensemble with independent complex Gaussian random variables. Based on this and the author’s earlier work in this direction, we obtain the Tracy–Widom law of the largest singular values of the Jacobi emsemble. Moreover, the circular law, the Marchenko–Pastur law, the central limit theorem, and the laws of large numbers for the spectral norms are also obtained.     

Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in <Emphasis Type="Italic">l</Emphasis><Superscript>2</Superscript> by the use of finite submatrices

We consider the problem of approximation of eigenvalues of a self-adjoint operator J defined by a Jacobi matrix in the Hilbert space l ²(ℕ) by eigenvalues of principal finite submatrices of an infinite Jacobi matrix that defines this operator. We assume the operator J is bounded from below with compact resolvent. In our research we estimate the asymptotics (with n → ∞) of the joint error of approximation for the eigenvalues, numbered from 1 to N; of J by the eigenvalues of the finite submatrix J _n of order n × n; where N = max{k ∈ ℕ: k ≤ rn} and r ∈ (0; 1) is arbitrary chosen. We apply this result to obtain an asymptotics for the eigenvalues of J. The method applied in this research is based on Volkmer’s results included in [23].     

Spectral theory of certain unbounded Jacobi matrices

Using the conjugate operator method of Mourre we study the spectral theory of a class of unbounded Jacobi matrices. We especially focus on the case where the off-diagonal entries an=nα(1+o(1)) and diagonal ones bn=λnα(1+o(1)) with α>0, λ∈R.     

On the eigenvalues of non-negative Jacobi matrices

The spectrum σ of a non-negative Jacobi matrix J is characterized. If J is also required to be irreducible, further conditions on σ are needed, some of which are explored.     

Bounds for eigenvalues of symmetric block Jacobi scaled matrices

The paper presents upper bounds for the largest eigenvalue of a block Jacobi scaled symmetric positive-definite matrix which depend only on such parameters as the block semibandwidth of a matrix and its block size. From these bounds we also derive upper bounds for the smallest eigenvalue of a symmetric matrix with identity diagonal blocks. Bibliography: 4 titles. Translated by L. Yu. Kolotilina. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 202, 1992, pp. 18–25.     

On the eigenvalues of principal submatrices of J-normal matrices

The problem of the existence of a J-normal matrix A when its spectrum and the spectrum of some of its (n-1)×(n-1) principal submatrices are prescribed is analyzed. The case of 3×3 matrices is particularly investigated. The results here obtained in the framework of indefinite inner product spaces are in the spirit of those due to Nylen, Tam and Uhlig.     

On the location of the eigenvalues of Jacobi matrices

Using some well known concepts on orthogonal polynomials, some recent results on the location of eigenvalues of tridiagonal matrices of very large order are extended. A significant number of important papers are unified.     

Inverse spectral theory of finite Jacobi matrices

We solve the following physically motivated problem: to determine all finite Jacobi matrices and corresponding indices such that the Green's function

is proportional to an arbitrary prescribed function . Our approach is via probability distributions and orthogonal polynomials.

We introduce what we call the auxiliary polynomial of a solution in order to factor the map

(where square brackets denote the equivalence class consisting of scalar multiples). This enables us to construct the solution set as a fibration over a connected, semi-algebraic coordinate base. The end result is a wealth of explicit constructions for Jacobi matrices. These reveal precise geometric information about the solution set, and provide the basis for new existence theorems.

     

Rank-deficient submatrices of Fourier matrices

We consider the maximal rank-deficient submatrices of Fourier matrices with order a power of a prime number. We do this by considering a hierarchical subdivision of these matrices into low rank blocks. We also explore some connections with the fast Fourier transform (FFT), and with an uncertainty principle for Fourier transforms over finite Abelian groups.     

m-Functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices

We study inverse spectral analysis for finite and semi-infinite Jacobi matricesH. Our results include a new proof of the central result of the inverse theory (that the spectral measure determinesH). We prove an extension of the theorem of Hochstadt (who proved the result in casen = N) thatn eigenvalues of anN × N Jacobi matrixH can replace the firstn matrix elements in determiningH uniquely. We completely solve the inverse problem for (δ _n , (H-z)^-1 δ _n ) in the caseN < ∞. This material is based upon work supported by the National Science Foundation under Grant Nos. DMS-9623121 and DMS-9401491.     

On the convergence of the classical Jacobi method for real symmetric matrices with non-distinct eigenvalues

Summary It is proved that the classical Jacobi method for real symmetric matrices with multiple eigenvalues converges quadratically.While this paper was in the press, SCHöNHAGE gave a different proof of the ultimate quadratic convergence of the classical Jacobi method. [See Numer. Math. 6, 410–412 (1964).     

On the eigenvalue problem for a particular class of finite Jacobi matrices

A function f with simple and nice algebraic properties is defined on a subset of the space of complex sequences. Some special functions are expressible in terms of f, first of all the Bessel functions of first kind. A compact formula in terms of the function f is given for the determinant of a Jacobi matrix. Further we focus on the particular class of Jacobi matrices of odd dimension whose parallels to the diagonal are constant and whose diagonal depends linearly on the index. A formula is derived for the characteristic function. Yet another formula is presented in which the characteristic function is expressed in terms of the function f in a simple and compact manner. A special basis is constructed in which the Jacobi matrix becomes a sum of a diagonal matrix and a rank-one matrix operator. A vector-valued function on the complex plain is constructed having the property that its values on spectral points of the Jacobi matrix are equal to corresponding eigenvectors.     

Embedded eigenvalues for perturbed periodic Jacobi operators using a geometric approach

We consider the problem of embedding eigenvalues into the essential spectrum of periodic Jacobi operators, using an oscillating, decreasing potential. To do this we employ a geometric method, previously used to embed eigenvalues into the essential spectrum of the discrete Schrödinger operator. For periodic Jacobi operators we relax the rational dependence conditions on the values of the quasi-momenta from this previous work. We then explore conditions that permit not just the existence of infinitely many subordinate solutions to the formal spectral equation but also the embedding of infinitely many eigenvalues.     

Properties of submatrices of Sylvester Hadamard matrices

We investigate the algebraic behaviour of leading principal submatrices of Hadamard matrices being powers of 2. We provide analytically the spectrum of general submatrices of these Hadamard matrices. Symmetry properties and relationships between the upper left and lower right corners of the matrices in this respect are demonstrated. Considering the specific construction scheme of this particular class of Hadamard matrices (called Sylvester Hadamard matrices), we utilize tensor operations to prove the respective results. An algorithmic procedure yielding the complete spectrum of leading principal submatrices of Sylvester Hadamard matrices is proposed.