scholarly journals A Formula for Eigenvalues of Jacobi Matrices with a Reflection Symmetry

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
S. B. Rutkevich
Keyword(s):  
Spectral Properties ◽  
Jacobi Operator ◽  
Scattering Data ◽  
Jacobi Matrices ◽  
Reflection Symmetry ◽  
Matrix Elements ◽  
Discrete Schrödinger Operator ◽  
Infinite Dimensional ◽  
Jacobi Operators ◽  
Secondary Diagonal

The spectral properties of two special classes of Jacobi operators are studied. For the first class represented by the 2M-dimensional real Jacobi matrices whose entries are symmetric with respect to the secondary diagonal, a new polynomial identity relating the eigenvalues of such matrices with their matrix entries is obtained. In the limit M→∞ this identity induces some requirements, which should satisfy the scattering data of the resulting infinite-dimensional Jacobi operator in the half-line, of which super- and subdiagonal matrix elements are equal to -1. We obtain such requirements in the simplest case of the discrete Schrödinger operator acting in l2(N), which does not have bound and semibound states and whose potential has a compact support.

scholarly journals Oscillation Theory for the Density of States of High Dimensional Random Operators

2017 ◽  
Vol 2019 (15) ◽  
pp. 4579-4602
Author(s):  
Julian Groß mann ◽  
Hermann Schulz-Baldes ◽  
Carlos Villegas-Blas
Keyword(s):  
Density Of States ◽  
Von Neumann Algebra ◽  
Jacobi Operator ◽  
Oscillation Theory ◽  
Integrated Density Of States ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Jacobi Operators ◽  
Sturm Liouville ◽  
Finite Trace

Abstract Sturm–Liouville oscillation theory is studied for Jacobi operators with block entries given by covariant operators on an infinite dimensional Hilbert space. It is shown that the integrated density of states of the Jacobi operator is approximated by the winding of the Prüfer phase w.r.t. the trace per unit volume. This rotation number can be interpreted as a spectral flow in a von Neumann algebra with finite trace.

scholarly journals On Spectral Properties of Compact Toeplitz Operators on Bergman Space with Logarithmically Decaying Symbol and Applications to Banded Matrices

2021 ◽  
Author(s):  
Mahamet Koïta ◽  
Stanislas Kupin ◽  
Sergey Naboko ◽  
Belco Touré
Keyword(s):  
Spectral Analysis ◽  
Continuous Function ◽  
Bergman Space ◽  
Orthogonal Projection ◽  
Spectral Properties ◽  
Toeplitz Operators ◽  
Closed Subspace ◽  
Jacobi Matrices ◽  
Banded Matrices ◽  
Eigen Values

Abstract Let $L^2({{\mathbb{D}}})$ be the space of measurable square-summable functions on the unit disk. Let $L^2_a({{\mathbb{D}}})$ be the Bergman space, that is, the (closed) subspace of analytic functions in $L^2({{\mathbb{D}}})$. $P_+$ stays for the orthogonal projection going from $L^2({{\mathbb{D}}})$ to $L^2_a({{\mathbb{D}}})$. For a function $\varphi \in L^\infty ({{\mathbb{D}}})$, the Toeplitz operator $T_\varphi : L^2_a({{\mathbb{D}}})\to L^2_a({{\mathbb{D}}})$ is defined as $$\begin{align*} & T_\varphi f=P_+\varphi f, \quad f\in L^2_a({{\mathbb{D}}}). \end{align*}$$The main result of this article are spectral asymptotics for singular (or eigen-) values of compact Toeplitz operators with logarithmically decaying symbols, that is, $$\begin{align*} & \varphi(z)=\varphi_1(e^{i\theta})\, (1+\log(1/(1-r)))^{-\gamma},\quad \gamma>0, \end{align*}$$where $z=re^{i\theta }$ and $\varphi _1$ is a continuous (or piece-wise continuous) function on the unit circle. The result is applied to the spectral analysis of banded (including Jacobi) matrices.

scholarly journals Spectral Properties of Block Jacobi Matrices

2018 ◽  
Vol 48 (2) ◽  
pp. 301-335 ◽  
Author(s):  
Grzegorz Świderski
Keyword(s):  
Spectral Properties ◽  
Jacobi Matrices

scholarly journals On unbounded commuting Jacobi operators and some related issues

2019 ◽  
Vol 6 (1) ◽  
pp. 82-91
Author(s):  
Andrey Osipov
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Jacobi Matrices ◽  
Unbounded Operators ◽  
Jacobi Operators ◽  
Necessary And Sufficient

Abstract We consider the situations, when two unbounded operators generated by infinite Jacobi matrices, are self-adjoint and commute. It is found that if two Jacobi matrices formally commute, then two corresponding operators are either self-adjoint and commute, or admit a commuting self-adjoint extensions. In the latter case such extensions are explicitly described. Also, some necessary and sufficient conditions for self-adjointness of Jacobi operators are studied.

scholarly journals A Special Class of Infinite Dimensional Dirac Operators on the Abstract Boson-Fermion Fock Space

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Asao Arai
Keyword(s):  
Dirac Operator ◽  
Hilbert Spaces ◽  
Special Class ◽  
Spectral Properties ◽  
Fock Space ◽  
Dirac Operators ◽  
Fredholm Property ◽  
Perturbation Parameter ◽  
Coupling Constant ◽  
Infinite Dimensional

Spectral properties of a special class of infinite dimensional Dirac operatorsQ(α)on the abstract boson-fermion Fock spaceℱ(ℋ,𝒦)associated with the pair(ℋ,𝒦)of complex Hilbert spaces are investigated, whereα∈Cis a perturbation parameter (a coupling constant in the context of physics) and the unperturbed operatorQ(0)is taken to be a free infinite dimensional Dirac operator. A variety of the kernel ofQ(α)is shown. It is proved that there are cases where, for all sufficiently large|α|withα<0,Q(α)has infinitely many nonzero eigenvalues even ifQ(0)has no nonzero eigenvalues. Also Fredholm property ofQ(α)restricted to a subspace ofℱ(ℋ,𝒦)is discussed.

scholarly journals Analytic Scattering Theory for Jacobi Operators and Bernstein–Szegö Asymptotics of Orthogonal Polynomials

2018 ◽  
Vol 30 (08) ◽  
pp. 1840019 ◽  
Author(s):  
D. R. Yafaev
Keyword(s):  
Jacobi Matrix ◽  
Trace Class ◽  
Spectral Shift ◽  
Shift Function ◽  
Discrete Schrödinger Operator ◽  
Wave Operators ◽  
Jacobi Operators ◽  
Matrix Formula ◽  
Asymptotics Of Orthogonal Polynomials ◽  
Szego Asymptotics

We study semi-infinite Jacobi matrices [Formula: see text] corresponding to trace class perturbations [Formula: see text] of the “free” discrete Schrödinger operator [Formula: see text]. Our goal is to construct various spectral quantities of the operator [Formula: see text], such as the weight function, eigenfunctions of its continuous spectrum, the wave operators for the pair [Formula: see text], [Formula: see text], the scattering matrix, the spectral shift function, etc. This allows us to find the asymptotic behavior of the orthonormal polynomials [Formula: see text] associated to the Jacobi matrix [Formula: see text] as [Formula: see text]. In particular, we consider the case of [Formula: see text] inside the spectrum [Formula: see text] of [Formula: see text] when this asymptotic has an oscillating character of the Bernstein–Szegö type and the case of [Formula: see text] at the end points [Formula: see text].

Electronic and Optical Properties of HgI2

1993 ◽  
Vol 302 ◽  
Author(s):  
Yia-Chung Chang ◽  
Hock-Kee Sim ◽  
R. B. James
Keyword(s):  
Electronic Structures ◽  
Electronic Band Structure ◽  
Scattering Data ◽  
Matrix Elements ◽  
Interband Transitions ◽  
Phonon Modes ◽  
Electronic Band ◽  
Optical Responses ◽  
Tetragonal Form ◽  
Excitonic Effects

ABSTRACTWe present theoretical studies of electronic structures, optical responses, and phonon modes of undoped HgI2 in its red tetragonal form. The electronic band structure is studied via an empirical nonlocal pseudopotential model, including the spin-orbit interaction. The electron and hole effective masses, optical matrix elements for interband transitions, and complex dielectric function are computed. Excitonic effects on the absorption coefficient near the fundamental band gap are included within the effectivemass approximation. The resulting absorption spectra and their polarization dependence are compared with experiment with favorable agreement. The phonon modes of HgI2 are studied with a microscopic model and a good fit to the neutron scattering data is obtained.

scholarly journals REAL HYPERSURFACES WITH CYCLIC-PARALLEL STRUCTURE JACOBI OPERATORS IN A NONFLAT COMPLEX SPACE FORM

2009 ◽  
Vol 81 (2) ◽  
pp. 260-273 ◽  
Author(s):  
U-HANG KI ◽  
HIROYUKI KURIHARA
Keyword(s):  
Complex Space ◽  
Space Form ◽  
Jacobi Operator ◽  
Complex Space Form ◽  
Parallel Structure ◽  
Real Hypersurfaces ◽  
Structure Jacobi Operator ◽  
Jacobi Operators

AbstractIt is known that there are no real hypersurfaces with parallel structure Jacobi operators in a nonflat complex space form. In this paper, we classify real hypersurfaces in a nonflat complex space form whose structure Jacobi operator is cyclic-parallel.

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