On infinite Jacobi matrices with a trace class resolvent

Isospectral flows on a class of finite-dimensional Jacobi matrices

We present a new matrix-valued isospectral ordinary differential equation that asymptotically block-diagonalizes n × n zero-diagonal Jacobi matrices employed as its initial condition. This o.d.e. features a right-hand side with a nested commutator of matrices, and structurally resembles the double-bracket o.d.e. studied by R.W. Brockett in 1991. We prove that its solutions converge asymptotical...

Seminar on Jacobi Matrices

Borg-type Theorems for Generalized Jacobi Matrices and Trace Formulas

The paper deals with two types of inverse spectral problems for the class of generalized Jacobi matrices introduced in [9]. Following the scheme proposed in [5], we deduce analogs of the Hochstadt–Lieberman theorem and the Borg theorem. Properties of a Weyl function of the generalized Jacobi matrix are systematically used to prove the uniqueness theorems. Trace formulas for the generalized Jaco...

Two modified Jacobi methods for M-matrices

On a Spectral Property of Jacobi Matrices

Let J be a Jacobi matrix with elements bk on the main diagonal and elements ak on the auxiliary ones. We suppose that J is a compact perturbation of the free Jacobi matrix. In this case the essential spectrum of J coincides with [−2, 2], and its discrete spectrum is a union of two sequences {xj }, x + j > 2, x − j < −2, tending to ±2. We denote sequences {ak+1 − ak} and {ak+1 + ak−1 − 2ak} by ∂...