Poisson kernel expansions for Schrödinger operators on trees

Nalini Anantharaman; Mostafa Sabri

doi:10.4171/JST/247

Poisson kernel expansions for Schrödinger operators on trees

Nalini Anantharaman, Mostafa Sabri

Science

Research output: Contribution to journal › Article › peer-review

Abstract

We study Schrödinger operators on trees and construct associated Poisson kernels, in analogy to the Laplacian on the unit disc. We show that in the absolutely continuous spectrum, the generalized eigenfunctions of the operator are generated by the Poisson kernel. We use this to define a “Fourier transform”, giving a Fourier inversion formula and a Plancherel formula, where the domain of integration runs over the energy parameter and the geometric boundary of the tree.

Original language	English (US)
Pages (from-to)	243-268
Number of pages	26
Journal	Journal of Spectral Theory
Volume	9
Issue number	1
DOIs	https://doi.org/10.4171/JST/247
State	Published - 2019

Keywords

Generalized eigenfunctions
Poisson kernel
Schrödinger operator
Trees

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics
Geometry and Topology

Access to Document

10.4171/JST/247

Cite this

@article{c1e066fb93e248a7b7efba6cc23a1b41,

title = "Poisson kernel expansions for Schr{\"o}dinger operators on trees",

abstract = "We study Schr{\"o}dinger operators on trees and construct associated Poisson kernels, in analogy to the Laplacian on the unit disc. We show that in the absolutely continuous spectrum, the generalized eigenfunctions of the operator are generated by the Poisson kernel. We use this to define a “Fourier transform”, giving a Fourier inversion formula and a Plancherel formula, where the domain of integration runs over the energy parameter and the geometric boundary of the tree.",

keywords = "Generalized eigenfunctions, Poisson kernel, Schr{\"o}dinger operator, Trees",

author = "Nalini Anantharaman and Mostafa Sabri",

note = "Funding Information: This material is based upon work supported by the Agence Nationale de la Recherche under grant No. ANR-13-BS01-0007-01, by the Labex IRMIA and the Institute of Advance Study of Universit{\'e} de Strasbourg, and by Institut Universitaire de France. Publisher Copyright: {\textcopyright} European Mathematical Society.",

year = "2019",

doi = "10.4171/JST/247",

language = "English (US)",

volume = "9",

pages = "243--268",

journal = "Journal of Spectral Theory",

issn = "1664-039X",

publisher = "European Mathematical Society Publishing House",

number = "1",

}

TY - JOUR

T1 - Poisson kernel expansions for Schrödinger operators on trees

AU - Anantharaman, Nalini

AU - Sabri, Mostafa

N1 - Funding Information: This material is based upon work supported by the Agence Nationale de la Recherche under grant No. ANR-13-BS01-0007-01, by the Labex IRMIA and the Institute of Advance Study of Université de Strasbourg, and by Institut Universitaire de France. Publisher Copyright: © European Mathematical Society.

PY - 2019

Y1 - 2019

N2 - We study Schrödinger operators on trees and construct associated Poisson kernels, in analogy to the Laplacian on the unit disc. We show that in the absolutely continuous spectrum, the generalized eigenfunctions of the operator are generated by the Poisson kernel. We use this to define a “Fourier transform”, giving a Fourier inversion formula and a Plancherel formula, where the domain of integration runs over the energy parameter and the geometric boundary of the tree.

AB - We study Schrödinger operators on trees and construct associated Poisson kernels, in analogy to the Laplacian on the unit disc. We show that in the absolutely continuous spectrum, the generalized eigenfunctions of the operator are generated by the Poisson kernel. We use this to define a “Fourier transform”, giving a Fourier inversion formula and a Plancherel formula, where the domain of integration runs over the energy parameter and the geometric boundary of the tree.

KW - Generalized eigenfunctions

KW - Poisson kernel

KW - Schrödinger operator

KW - Trees

UR - http://www.scopus.com/inward/record.url?scp=85061344870&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85061344870&partnerID=8YFLogxK

U2 - 10.4171/JST/247

DO - 10.4171/JST/247

M3 - Article

AN - SCOPUS:85061344870

SN - 1664-039X

VL - 9

SP - 243

EP - 268

JO - Journal of Spectral Theory

JF - Journal of Spectral Theory

IS - 1

ER -

Poisson kernel expansions for Schrödinger operators on trees

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this