Resolvent expansion for the Schrödinger operator on a graph with infinite rays

Kenichi Ito, Arne Jensen

Research output: Contribution to journalJournal articleResearchpeer-review

Abstract

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.
Original languageEnglish
JournalJournal of Mathematical Analysis and Applications
Volume464
Issue number1
Pages (from-to)616-661
Number of pages46
ISSN0022-247X
DOIs
Publication statusPublished - Aug 2018

Keywords

  • Schrödinger operator
  • Threshold
  • Resonance
  • Generalized eigenfunction
  • Resolvent expansion
  • Combinatorial graph

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