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

Fingerprint

Resolvent
Eigenfunctions
Half line
Eigenvalues and eigenfunctions
Zero
Finite Graph
Graph in graph theory
Operator
Justify
Asymptotic Expansion
Simplify
Eigenvalue
Coefficient

Keywords

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

Cite this

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title = "Resolvent expansion for the Schr{\"o}dinger operator on a graph with infinite rays",
abstract = "We consider the Schr{\"o}dinger 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.",
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Resolvent expansion for the Schrödinger operator on a graph with infinite rays. / Ito, Kenichi; Jensen, Arne.

In: Journal of Mathematical Analysis and Applications, Vol. 464, No. 1, 08.2018, p. 616-661.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Resolvent expansion for the Schrödinger operator on a graph with infinite rays

AU - Ito, Kenichi

AU - Jensen, Arne

PY - 2018/8

Y1 - 2018/8

N2 - 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.

AB - 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.

KW - Schrödinger operator

KW - Threshold

KW - Resonance

KW - Generalized eigenfunction

KW - Resolvent expansion

KW - Combinatorial graph

U2 - 10.1016/j.jmaa.2018.04.022

DO - 10.1016/j.jmaa.2018.04.022

M3 - Journal article

VL - 464

SP - 616

EP - 661

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -