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Ambarzumian-type mixed inverse spectral problems for Jacobi matrices

Ethan Luo, Steven Ning, Tarun Rapaka, Xuxuan Joyce Zheng

TL;DR

This paper studies Ambarzumian-type mixed inverse spectral problems for Jacobi matrices, focusing on the finite tridiagonal matrix with missing diagonal entries and the free discrete Schrödinger matrix $F_n$. It defines $S_{n,m}$ and investigates whether sharing $m$ consecutive eigenvalues (Q1) or $m$ ordered eigenvalues (Q2) with $F_n$ enforces $S_{n,m}=F_n$; the analysis centers on $(n,m)=(5,3)$. Through Macaulay2 computations and hand-case analysis, the authors show that Q1 holds for this case (yielding $S_{5,3}=F_5$ when eigenvalues coincide as described) but Q2 can fail: Case 4 yields numerical nontrivial solutions with the same ordered eigenvalues yet $S_{5,3} eq F_5$. The work provides computational insight into inverse spectral rigidity for small matrices and outlines a program to prove the counterexample rigorously and extend the results to larger sizes and nonconsecutive eigenvalue information.

Abstract

We investigate Ambarzumian-type mixed inverse spectral problems for Jacobi matrices. Specifically, we examine whether the Jacobi matrix can be uniquely determined by knowing all but the first $m$ diagonal entries and a set of $m$ ordered eigenvalues.

Ambarzumian-type mixed inverse spectral problems for Jacobi matrices

TL;DR

This paper studies Ambarzumian-type mixed inverse spectral problems for Jacobi matrices, focusing on the finite tridiagonal matrix with missing diagonal entries and the free discrete Schrödinger matrix . It defines and investigates whether sharing consecutive eigenvalues (Q1) or ordered eigenvalues (Q2) with enforces ; the analysis centers on . Through Macaulay2 computations and hand-case analysis, the authors show that Q1 holds for this case (yielding when eigenvalues coincide as described) but Q2 can fail: Case 4 yields numerical nontrivial solutions with the same ordered eigenvalues yet . The work provides computational insight into inverse spectral rigidity for small matrices and outlines a program to prove the counterexample rigorously and extend the results to larger sizes and nonconsecutive eigenvalue information.

Abstract

We investigate Ambarzumian-type mixed inverse spectral problems for Jacobi matrices. Specifically, we examine whether the Jacobi matrix can be uniquely determined by knowing all but the first diagonal entries and a set of ordered eigenvalues.
Paper Structure (8 sections, 22 equations)

This paper contains 8 sections, 22 equations.

Table of Contents

  1. Introduction
  2. Macaulay2
  3. Specific cases checked by hand
  4. Case 1: $\mathbf{\widetilde{\lambda_2} = \lambda_2 = -1}$, $\mathbf{\widetilde{\lambda_3} = \lambda_3 = 0}$, and $\mathbf{\widetilde{\lambda_4} = \lambda_4 = 1}$.
  5. Case 2: $\mathbf{\widetilde{\lambda_1} = \lambda_1 = -\sqrt{3}}$, $\mathbf{\widetilde{\lambda_2} = \lambda_2 = -1}$, and $\mathbf{\widetilde{\lambda_4} = \lambda_4 = 1}$.
  6. Case 3: $\mathbf{\widetilde{\lambda_2} = \lambda_2 = -1}$, $\mathbf{\widetilde{\lambda_4} = \lambda_4 = 1}$, and $\mathbf{\widetilde{\lambda_5} = \lambda_5 = \sqrt{3}}$.
  7. Case 4: $\mathbf{\widetilde{\lambda_1} = \lambda_1 = -\sqrt{3}}$, $\mathbf{\widetilde{\lambda_3} = \lambda_3 = 0}$, and $\mathbf{\widetilde{\lambda_4} = \lambda_4 = 1}$
  8. Future work