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Spectral analysis of infinite Marchenko-Slavin matrices

Sergio Palafox, Luis O. Silva

Abstract

This work tackles the problem of spectral characterization of a class of infinite matrices arising from the modelling of small oscillations in a system of interacting particles. The class of matrices under discussion corresponds to the infinite Marchenko-Slavin class. The spectral functions of these matrices are completely characterized and an algorithm is provided for the reconstruction of the matrix from its spectral function. The techniques used in this work are based on recent results for the spectral characterization of infinite band symmetric matrices with so-called degenerations.

Spectral analysis of infinite Marchenko-Slavin matrices

Abstract

This work tackles the problem of spectral characterization of a class of infinite matrices arising from the modelling of small oscillations in a system of interacting particles. The class of matrices under discussion corresponds to the infinite Marchenko-Slavin class. The spectral functions of these matrices are completely characterized and an algorithm is provided for the reconstruction of the matrix from its spectral function. The techniques used in this work are based on recent results for the spectral characterization of infinite band symmetric matrices with so-called degenerations.

Paper Structure

This paper contains 6 sections, 25 theorems, 62 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The class $\mathfrak M$ of semi-infinite matrices
  3. Spectral functions of submatrices
  4. Band diagonal matrices with degenerations
  5. Spectral functions for the class $\mathfrak M$
  6. Reconstruction of the matrix

Key Result

Lemma 3.1

For any $z\in\mathbb{C}$, the $N$-dimensional vector $\eta(z)$ is a solution to eq:formal-difference_p if and only if $\eta(z)=\Psi_{\mathscr T}(z) C$, where $C$ is an $n$-dimensional vector. If $C_{1}$ and $C_{2}$ are two linearly independent vectors in $l_{2}(G_{n})$, then $\Psi_{\mathscr T}(z) C_

Figures (4)

  • Figure 1: An example of an element of $\mathfrak{M}$
  • Figure 2: A submatrix of $\mathfrak{M}$ with $N=7$ and $n=3$.
  • Figure 3: The structure of a matrix in $\widetilde{\mathfrak M}$
  • Figure 4: Orthonormalization algorithm

Theorems & Definitions (59)

  • Definition 1
  • Definition 2
  • Example 1
  • Lemma 3.1
  • proof
  • Proposition 3.1
  • proof
  • Corollary 3.1
  • proof
  • Lemma 3.2
  • ...and 49 more