2017/12/05 by Kenichi Ito, Arne Jensen, Ito, Kenichi +1
Computer Science · Mathematics · Physics and Astronomy · #47A10 #Asymptotic expansion #Discrete mathematics #Eigenfunction #Eigenvalues and eigenvectors #FOS: Mathematics #FOS: Physical sciences #Finite-rank operator #Graph #Graph theory and applications #Mathematical Physics (math-ph) #Mathematical analysis #Mathematics #Matrix Theory and Algorithms #Operator (biology) #Physics #Pure mathematics #Quantum mechanics #Resolvent #Resolvent formalism #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.SP #msc:47A10
paper · pdf · doi:10.48550/arxiv.1712.01592
published in arXiv (Cornell University) (Cornell University) · 55 pages, minor revisions, final version
openalex publication_date 2017/12/05 · arxiv created 2018/04/16 · arxiv updated 2018/04/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We consider the Schrödinger operator on a combinatorial graph consisting of a finite graph and a finite number of discrete half-lines, all jointed together, and compute an asymptotic expansion of its resolvent around the threshold 0. Precise expressions are obtained for the first few coefficients of the expansion in terms of the generalized eigenfunctions. This result justifies the classification of threshold types solely by growth properties of the generalized eigenfunctions. By choosing an appropriate free operator a priori possessing no zero eigenvalue or zero resonance we can simplify the expansion procedure as much as that on the single discrete half-line.