Inverse Spectral Problems for Tridiagonal N by N Complex Hamiltonians

TL;DR

This paper introduces a new approach to inverse spectral problems for complex tridiagonal matrices, providing methods to reconstruct such matrices from spectral data, with applications to matrices with real eigenvalues.

Contribution

It develops a framework for inverse spectral problems using generalized spectral functions for complex Jacobi matrices, including a construction procedure for matrices with real eigenvalues.

Findings

Established a structure for generalized spectral functions of complex Jacobi matrices.

Developed a reconstruction procedure for complex tridiagonal matrices from spectral data.

Provided methods to construct matrices with real eigenvalues from spectral information.

Abstract

In this paper, the concept of generalized spectral function is introduced for finite-order tridiagonal symmetric matrices (Jacobi matrices) with complex entries. The structure of the generalized spectral function is described in terms of spectral data consisting of the eigenvalues and normalizing numbers of the matrix. The inverse problems from generalized spectral function as well as from spectral data are investigated. In this way, a procedure for construction of complex tridiagonal matrices having real eigenvalues is obtained.