Related papers: Persymmetric Jacobi matrices with square-integer e…
The circular and Jacobi ensembles of random matrices have their eigenvalue support on the unit circle of the complex plane and the interval $(0,1)$ of the real line respectively. The averaged value of the modulus of the corresponding…
The classical theory of Kosambi-Cartan-Chern (KCC) developed in differential geometry provides a powerful method for analyzing the behaviors of dynamical systems. In the KCC theory, the properties of a dynamical system are described in…
We construct an ensemble of correlation matrices from high-frequency foreign exchange market data, with one matrix for every day for 446 days. The matrices are symmetric and have vanishing diagonal elements after subtracting the identity…
We investigate how invariant subspaces corresponding to a single eigenvalue will change when a matrix is perturbed. We focus on the invariant subspaces corresponding to an eigenvalue associated with the Jordan blocks that have the same…
We describe a suite of fast algorithms for evaluating Jacobi polynomials, applying the corresponding discrete Sturm-Liouville eigentransforms and calculating Gauss-Jacobi quadrature rules. Our approach is based on the well-known fact that…
A Bernstein type inequality is obtained for the Jacobi polynomials $P_n^{\alpha,\beta}(x)$, which is uniform for all degrees $n\ge0$, all real $\alpha,\beta\ge0$, and all values $x\in [-1,1]$. It provides uniform bounds on a complete set of…
In this work the method of analyzing of the absolutely continuous spectrum for self-adjoint operators is considered. For the analysis it is used an approximation of self-adjoint operator $A$ by a sequence of operators $A_n$ with absolutely…
Three ways of constructing a non-Hermitian matrix with possible all real eigenvalues are discussed. They are PT symmetry, pseudo-Hermiticity, and generalized PT symmetry. Parameter counting is provided for each class. All three classes of…
We study the bispectrality of Jacobi type polynomials, which are eigenfunctions of higher-order differential operators and can be defined by taking suitable linear combinations of a fixed number of consecutive Jacobi polynomials. Jacobi…
In this paper we study various difference equations related to Jacobi-type pencils. By a Jacobi-type pencil one means the following pencil: $J_5 - \lambda J_3$, where $J_3$ is a Jacobi matrix and $J_5$ is a semi-infinite real symmetric…
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…
In general, point spectrum of an almost periodic Jacobi matrix can depend on the element of the hull. In this paper, we study the hull of the limit-periodic Jacobi matrix corresponding to the equilibrium measure of the Julia set of the…
In this paper, we study the gap probability problem of the (symmetric) Jacobi unitary ensemble of Hermitian random matrices, namely the probability that the interval $(-a,a)\:(0<a<1)$ is free of eigenvalues. Using the ladder operator…
The paper deals with the distribution of singular values of the input-output Jacobian of deep untrained neural networks in the limit of their infinite width. The Jacobian is the product of random matrices where the independent rectangular…
We investigate the large $n$ asymptotics of the $n$-th eigenvalue for a class of unbounded self-adjoint operators defined by infinite Jacobi matrices with discrete spectrum. In the case of the quantum Rabi model we obtain the first three…
The partly symmetric real Ginibre ensemble consists of matrices formed as linear combinations of real symmetric and real anti-symmetric Gaussian random matrices. Such matrices typically have both real and complex eigenvalues. For a fixed…
We study the spectral properties of bounded and unbounded Jacobi matrices whose entries are bounded operators on a complex Hilbert space. In particular, we formulate conditions assuring that the spectrum of the studied operators is…
Pseudo-hermitian matrices are matrices hermitian with respect to an indefinite metric. They can be thought of as the truncation of pseudo-hermitian operators, defined over some Krein space, together with the associated metric, to a finite…
Classical Jacobi polynomials $P_{n}^{(\alpha,\beta)}$, with $\alpha, \beta>-1$, have a number of well-known properties, in particular the location of their zeros in the open interval $(-1,1)$. This property is no longer valid for other…
The spectrum of three-diagonal self-adjoint $p$-periodic Jacobi matrix with positive off-diagonal elements $a_n$ an real diagonal elements $b_n$ consist of intervals separated by $p-1$ gaps $\gamma_i$, where some of the gaps can be…