Inverse spectral analysis for finite matrix-valued Jacobi operators

Abstract

Consider the Jacobi operators $\cJ$ given by $(\cJ y)_n=a_ny_{n+1}+b_ny_n+a_{n-1}^*y_{n-1}$, $y_n\in \C^m$ (here $y_0=y_{p+1}=0$), where $b_n=b_n^{\ast }$ and $a_n:\det a_n\ne 0$ are the sequences of $m\ts m$ matrices, $n=1,..,p$. We study two cases: (i) $a_n=a_n^{\ast }>0$; (ii) $a_n$ is a lower triangular matrix with real positive entries on the diagonal (the matrix $\cJ$ is $(2m+1)$-band $mp\ts mp$ matrix with positive entries on the first and the last diagonals). The spectrum of $\cJ$ is a finite sequence of real eigenvalues ${ł}_1<...<{ł}_N$, where each eigenvalue ${ł}_j$ has multiplicity $k_j\le m$. We show that the mapping $(a,b)\mapsto \{({ł}_j,k_j){\}}_1^N\oplus \{additionalspectraldata\}$ is 1-to-1 and onto. In both cases (i), \nolinebreak (ii), we give the complete solution of the inverse problem.

Cited by in corpus (1)

• Determinants of Block Tridiagonal Matrices