scholarly journals Spectral analysis of a class of hermitian Jacobi matrices in a critical (double root) hyperbolic case

2010 ◽  
Vol 53 (1) ◽  
pp. 239-254 ◽  
Author(s):  
Serguei Naboko ◽  
Sergey Simonov
Keyword(s):  
Spectral Analysis ◽  
Jacobi Matrices ◽  
Double Root ◽  
Full Generality ◽  
Hyperbolic Case ◽  
Generalized Eigenvectors

AbstractWe 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 analyse the spectrum. In addition, we reformulate a very helpful theorem from a paper by Janas and Moszynski in its full generality in order to serve the needs of our method.

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 analysis of two doubly infinite Jacobi matrices with exponential entries

2019 ◽  
Vol 276 (6) ◽  
pp. 1681-1716
Author(s):  
Mourad E.H. Ismail ◽  
František Štampach
Keyword(s):  
Spectral Analysis ◽  
Jacobi Matrices

Generalized Jacobi Matrices and Spectral Analysis of Differential Operators with Polynomial Coefficients

2020 ◽  
Vol 252 (2) ◽  
pp. 213-224
Author(s):  
K. A. Mirzoev ◽  
N. N. Konechnaya ◽  
T. A. Safonova ◽  
R. N. Tagirova
Keyword(s):  
Spectral Analysis ◽  
Differential Operators ◽  
Jacobi Matrices ◽  
Polynomial Coefficients
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