On the spectrum of Schrödinger operators with a finitely supported potential on the \(d\)-regular tree
type: Item
instance of: scholarly article
title: On the spectrum of Schrödinger operators with a finitely supported potential on the \(d\)-regular tree
zbMATH Open document ID: 1187.39002
author: Kazunori Ando
author: Yusuke Higuchi
published in: Linear Algebra and its Applications
publication date: 2009-10-09T00:00:00Z
review text: The authors characterize the spectrum of the discrete Schrödinger operator \[ Hf(x)=V(x)f(x)+\frac{1}{d}\sum_{e\in A_x} [f(t(e))-f(o(e))] \] in \(l^2(T_d)\), where \(T_d\) is a \(d\)-regular tree, \(A_x\) is the set of all oriented edges of \(T_d\) originating at the vertex \(x\), and \(t(e)\) (resp. \(o(e)\)) denote the terminus (resp. origin) of the oriented edge \(e\). Fixing \(v\in \mathbb{R}\) and \(l\) a non-negative integer, for each vertex \(x\) the potential \(V(x)\) is assumed to \(1+v\) for \(d(o,x)=l\) and \(1\) otherwise, where \(o\) is the origin of \(T_d\) and \(d(o,x)\) is the graph distance from \(o\) to \(x\). Denoting the vertex set of \(T_d\) by \({\mathcal V}\), the Hilbert space \(l^2(T_d)\) is given by \[ l^2(T_d) =\left\{f:{\mathcal V}\to \mathbb{C}\;\left|\;\sum_{x\in{\mathcal V}} \frac{1}{d} |f(x)|^2<\infty\right\}\right.. \]
reviewed by: Bruce A. Watson
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 47B39
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 39A70
zbMATH DE Number: 5612569
zbMATH Keywords: graphs
zbMATH Keywords: trees
zbMATH Keywords: spectrum
zbMATH Keywords: discrete Laplacian
zbMATH Keywords: discrete Schrödinger operators
MaRDI profile type: Publication
full work available at URL: j.laa.2009.06.033
OpenAlex ID: W1975440127
cites work: Q3414405
cites work: Stability of the absolutely continuous spectrum of random Schrödinger operators on tree graphs
cites work: A Mourre estimate for a Schrödinger operator on a binary tree.
cites work: Q4539490
cites work: Singular continuous spectrum for the Laplacian on certain sparse trees
cites work: Variational estimates for discrete Schrödinger operators with potentials of indefinite sign
cites work: Bound states and the Szegő condition for Jacobi matrices and Schrödinger operators.
cites work: Schrödinger operators with few bound states
cites work: Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schrödinger operators on graphs
cites work: Q5526485
cites work: Q3836536
DOI: 10.1016/J.LAA.2009.06.033
Recommended article: On the essential spectrum of Schrödinger operators on trees
Recommended article: The discrete spectrum of Schrödinger operators with \(\delta\)-type conditions on regular metric trees
Recommended article: Discreteness of spectrum for Schrödinger operators with δʹ-type conditions on infinite regular trees
Recommended article: Spectral properties of Schrödinger operators on radial \(N\)-dimensional infinite trees
Recommended article: Spectrum of Schrödinger operator on a homogeneous tree
Recommended article: Absolutely continuous spectrum of a Schrödinger operator on a tree
Recommended article: Q4431872
Recommended article: SCHRÖDINGER OPERATORS ON HOMOGENEOUS METRIC TREES: SPECTRUM IN GAPS
Recommended article: On the spectrums of ergodic Schrodinger operators with finitely valued potentials
Recommended article: On the absolutely continuous spectrum of Sturm-Liouville operators with applications to radial quantum trees
description: scientific article; zbMATH DE number 5612569