Inverse eigenvalue problem for discrete Schrödinger operators of a graph

2025/06/18 by Anzila Laikhuram, Laikhuram, Anzila, Jephian C.-H. Lin +1
Mathematics · #05C50 #15A18 #15B57 #65F18 #Combinatorics (math.CO) #Differential Equations and Boundary Problems #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2506.15430

openalex publication_date 2025/06/18 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28

Abstract

A discrete Schrödinger operator of a graph G is a real symmetric matrix whose i,j-entry, i ≠ j, is negative if \i,j\ is an edge and zero if it is not an edge, while diagonal entries can be any real numbers. The discrete Schrödinger operators have been used to study vibration theory and the Colin de Verdière parameter. The inverse eigenvalue problem for discrete Schrödinger operators of a graph aims to characterize the possible spectra among discrete Schrödinger operators of a graph. Compared to the inverse eigenvalue problem of a graph, the answers turn out to be more limited, and several restrictions based on graph structure are given. Using the strong properties, analogous versions of the supergraph lemma, the liberation lemma, and the bifurcation lemma are established. Using these results, the inverse eigenvalue problem for discrete Schrödinger operators is resolved for each graph with at most 5 vertices.

Citations

• Inverse eigenvalue problem for Laplacian matrices of a graph
• The inverse nullity pair problem and the strong nullity interlacing property
• The liberation set in the inverse eigenvalue problem of a graph
• The bifurcation lemma for strong properties in the inverse eigenvalue problem of a graph
• Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra
• The strong spectral property for graphs
• The inverse eigenvalue problem of a graph: Multiplicities and minors
• Generalizations of the Strong Arnold Property and the minimum number of\n distinct eigenvalues of a graph

Related

• The inverse eigenvalue problem of a graph: Multiplicities and minors
• The liberation set in the inverse eigenvalue problem of a graph
• The bifurcation lemma for strong properties in the inverse eigenvalue problem of a graph
• Inverse spectral problem for the Schrödinger operator on the square lattice
• Schrödinger operators on periodic discrete graphs