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

Kenichi Ito, Arne Jensen

Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › peer review

Resumé

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.
OriginalsprogEngelsk
TidsskriftJournal of Mathematical Analysis and Applications
Vol/bind464
Udgave nummer1
Sider (fra-til)616-661
Antal sider46
ISSN0022-247X
DOI
StatusUdgivet - aug. 2018

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

Citer dette

Ito, Kenichi ; Jensen, Arne. / Resolvent expansion for the Schrödinger operator on a graph with infinite rays. I: Journal of Mathematical Analysis and Applications. 2018 ; Bind 464, Nr. 1. s. 616-661.
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Ito, K & Jensen, A 2018, 'Resolvent expansion for the Schrödinger operator on a graph with infinite rays', Journal of Mathematical Analysis and Applications, bind 464, nr. 1, s. 616-661. https://doi.org/10.1016/j.jmaa.2018.04.022

Resolvent expansion for the Schrödinger operator on a graph with infinite rays. / Ito, Kenichi; Jensen, Arne.

I: Journal of Mathematical Analysis and Applications, Bind 464, Nr. 1, 08.2018, s. 616-661.

Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › peer 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

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Ito K, Jensen A. Resolvent expansion for the Schrödinger operator on a graph with infinite rays. Journal of Mathematical Analysis and Applications. 2018 aug;464(1):616-661. https://doi.org/10.1016/j.jmaa.2018.04.022

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