Related papers: Jacobi matrices on trees
We study contact structures on nonnegatively-graded manifolds equipped with homological contact vector fields. In the degree 1 case, we show that there is a one-to-one correspondence between such structures (with fixed contact form) and…
A real persymmetric Jacobi matrix of order $n$ whose eigenvalues are $2k^2$ $(k=0, ..., n-1)$ is presented, with entries given as explicit functions of $n$. Besides the possible use for testing forward and inverse numerical algorithms, such…
We study infinite tree and ultrametric matrices, and their action on the boundary of the tree. For each tree matrix we show the existence of a symmetric random walk associated to it and we study its Green potential. We provide a…
We introduce a class of doubly infinite complex Jacobi matrices determined by a simple convergence condition imposed on the diagonal and off-diagonal sequences. For each Jacobi matrix belonging to this class, an analytic function, called a…
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…
We study spectral properties of bounded and unbounded complex Jacobi matrices. In particular, we formulate conditions assuring that the spectrum of the studied operators is continuous on some subsets of the complex plane and we provide…
We consider certain matrix-products where successive matrices in the product belong alternately to a particular qualitative class or its transpose. The main theorems relate structural and spectral properties of these matrix-products to the…
In this paper we introduce the notion of infinite dimensional Jacobi structure to describe the geometrical structure of a class of nonlocal Hamiltonian systems which appear naturally when applying reciprocal transformations to Hamiltonian…
A selfadjoined block tridiagonal matrix with positive definite blocks on the off-diagonals is by definition a Jacobi matrix with matrix entries. Transfer matrix techniques are extended in order to develop a rotation number calculation for…
In this paper, we develop holomorphic Jacobi structures. Holomorphic Jacobi manifolds are in one-to-one correspondence with certain homogeneous holomorphic Poisson manifolds. Furthermore, holomorphic Poisson manifolds can be looked at as…
A Jacobi matrix with matrix entries is a self-adjoint block tridiagonal matrix with invertible blocks on the off-diagonals. Averaging over boundary conditions leads to explicit formulas for the averaged spectral measure which can…
We introduce a class of Jacobi operators with discrete spectra which is characterized by a simple convergence condition. With any operator J from this class we associate a characteristic function as an analytic function on a suitable…
We obtain the asymptotic distribution of eigenvalues of real symmetric tridiagonal matrices as their dimension increases to infinity and whose diagonal and off-diagonal elements asymptotically change with the index n as J_{nt+i nt+i}\sim…
The Jacobian Conjecture would follow if it were known that real polynomial maps with a unipotent Jacobian matrix are injective. The conjecture that this is true even for $C^1$ maps is explored here. Some results known in the polynomial case…
It is known that for a totally positive (TP) matrix, the eigenvalues are positive and distinct and the eigenvector associated with the smallest eigenvalue is totally nonzero and has an alternating sign pattern. Here, a certain weakening of…
We are interested in the phenomenon of the essential spectrum instability for a class of unbounded (block) Jacobi matrices. We give a series of sufficient conditions for the matrices from certain classes to have a discrete spectrum on a…
There are several common ways to encode a tree as a matrix, such as the adjacency matrix, the Laplacian matrix (that is, the infinitesimal generator of the natural random walk), and the matrix of pairwise distances between leaves. Such…
Orthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We…
We introduce symmetric Poisson structures as pairs consisting of a symmetric bivector field and a torsion-free connection satisfying an integrability condition analogous to that in usual Poisson geometry. Equivalent conditions in Poisson…
We study resonances for Jacobi operators on the half lattice with matrix valued coefficient and finitely supported perturbations. We describe a forbidden domain, the geometry of resonances and their asymptotics when the main coefficient of…