On the eigenvalue problem for a particular class of finite Jacobi matrices

TL;DR

This paper introduces a special function to analyze the eigenvalues of a specific class of finite Jacobi matrices, providing explicit formulas for determinants and characteristic functions, and constructing eigenvectors in a novel basis.

Contribution

It defines a new algebraic function to express determinants and characteristic functions of certain Jacobi matrices, offering explicit formulas and a basis for eigenvector construction.

Findings

Derived a compact formula for the determinant of Jacobi matrices using the function .

Presented a simple formula for the characteristic function in terms of .

Constructed a basis where the Jacobi matrix decomposes into a diagonal and rank-one matrix.

Abstract

A function $\mathfrak{F}$ with simple and nice algebraic properties is defined on a subset of the space of complex sequences. Some special functions are expressible in terms of $\mathfrak{F}$, first of all the Bessel functions of first kind. A compact formula in terms of the function $\mathfrak{F}$ is given for the determinant of a Jacobi matrix. Further we focus on the particular class of Jacobi matrices of odd dimension whose parallels to the diagonal are constant and whose diagonal depends linearly on the index. A formula is derived for the characteristic function. Yet another formula is presented in which the characteristic function is expressed in terms of the function $\mathfrak{F}$ in a simple and compact manner. A special basis is constructed in which the Jacobi matrix becomes a sum of a diagonal matrix and a rank-one matrix operator. A vector-valued function on the complex plain is constructed having the property that its values on spectral points of the Jacobi matrix are equal to corresponding eigenvectors.