Spectrum of Schrödinger operator on a homogeneous tree
type: Item
instance of: scholarly article
title: Spectrum of Schrödinger operator on a homogeneous tree
zbMATH Open document ID: 0574.60069
author: A. G. Golitsyna
published in: Moscow University Mathematics Bulletin
publication date: 1985-01-01T00:00:00Z
review text: Let \(B_{\nu}\) be the Bethe tree (infinite Cayley tree), i.e. connected non-oriented tree, and in every of its vertices g, excluding one vertex \(g_ 0\) (root), \(\nu\) sides meet. The root is the origin of unique side. Denote by \(| g|\) the minimal number of sides in the path connecting g and \(g_ 0\) and for a finite set \(S\subset B_{\nu}\), \(| s| =card S\), consider a space \(\Phi_ S\) of the functions f(g), \(g\in S\) with the scalar product \((f,h)=\sum_{g\in S}f(g)\overline{h(g)}\). Define the selfadjoint operator \(H_ s^{\omega}\) as follows: \[ (H_ s^{\omega}f)(g)=\sum_{| g- g'| =1}a(g,g')f(g')+\xi (g)f(g) \] where \(a(g,g')=0\) if \(g\in S\), g'\(\in S\), and \(=\nu^{-1}\) if g,g'\(\in S\), \(a(g,g')=a(g',g)\) and \(\xi\) (g), \(g\in B_{\nu}\), are i.i.d. random variables. Let \(N_ s^{\omega}(\lambda)\) be the number of eigenvalues of \(H_ s^{\omega}\) not exceeding \(\lambda\), \(V_ n=\{g:| g| \leq n\}.\) Theorem 1. With probability 1 the limit (1) \(\lim_{n\to \infty}| V_ n|^{-1}N^{\omega}_{V_ n}(\lambda)\equiv N(\lambda)\) exists and is nonrandom. Theorem 1'. There exists the limit (1) for \(\xi\) \(\equiv 0\) and is purely discontinuous function with dense set of jumps. The proof of Th. 1' follows from straightforward calculations. The proof of Th. 1 uses the scheme proposed by the reviewer [Teor. Mat. Fiz. 6, 415-424 (1971; Zbl 0208.129)] and is based on the representation of the Laplace transform of \(N^{\omega}_{V_ n}(\lambda)\) as the expectation of the functional \(\exp\{\int^{+}_{0}\xi (g\tau)d\tau \}\) with respect to the continuous time random value \(g_ t\) on the tree \(B_{\nu}\) (the analog of the Feynman-Kac formula).
Mathematics Subject Classification ID: 60H25
Mathematics Subject Classification ID: 47B39
Mathematics Subject Classification ID: 39A70
zbMATH DE Number: 3917390
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: random finite-difference operator
zbMATH Keywords: integrated density of states
zbMATH Keywords: Bethe tree
zbMATH Keywords: Laplace transform
zbMATH Keywords: Feynman-Kac formula
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3917390