Spectral properties of quasiperiodic Kronig-Penney model
type: Item
instance of: scholarly article
title: Spectral properties of quasiperiodic Kronig-Penney model
zbMATH Open document ID: 1026.81014
author: Masahiro Kaminaga
author: Fumihiko Nakano
published in: Tsukuba Journal of Mathematics
publication date: 2003-05-19T00:00:00Z
review text: Let \(H_\theta\) be the self-adjoint operator on \(L^2(\mathbb{R})\), given by \[ H_\theta:= -{d^2\over dx^2}+ \sum_{j\in\mathbb{Z}} V_\theta(j)\delta(x-j), \] where \(\delta\) denotes the one-dimensional delta-function, and \(\{V_\theta(j)\}_{j\in\mathbb{Z}}\) is a sequence which takes only two positive values \(\lambda_1\neq \lambda_2\): \[ V_\theta(j):= \lambda_1\psi_A(\Phi(\alpha j)+\theta)+ \lambda_2\psi_{A^c}(\Phi(\alpha j)+ \theta),\quad j\in\mathbb{Z}. \] Here \(\chi_A\) is the characteristic function of a subset \(A\subset \mathbb{R}/\mathbb{Z}\), \(\Phi\) is the canonical projection from \(\mathbb{R}\) onto \(\mathbb{R}/\mathbb{Z}\), \(\alpha\in (0,1)\) and \(\theta\in \mathbb{R}/\mathbb{Z}\). The authors prove the absence of the point spectrum of \(H_\theta\) under certain conditions and show that in the case of J\(A= \Phi([1- \alpha, 1])\) and \(\alpha\) irrational, \(\sigma(H_\theta)\) is independent of \(\theta\). Moreover, \(\sigma(H_\theta)\) can be represented in terms of \(\sigma(h_0(\lambda))\), where \(h_\theta(\lambda)\) is a discrete analogue of \(H_\theta\): \[ (h_\theta(\lambda) u)(n):= u(n+1)+ u(n- 1)+ \lambda v_\theta(n)u(n), \qquad v_\theta(n)= \chi_A(\Phi(\alpha n)+\theta), \] and \(\lambda\in C\), where \(C\) is a certain subset of \(\mathbb{R}\).
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 81V70
Mathematics Subject Classification ID: 34L40
zbMATH DE Number: 1880960
zbMATH Keywords: selfadjoint operator
zbMATH Keywords: absence of point spectrum
reviewed by: Viorel Iftimie
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496164421
OpenAlex ID: W1488045715
DOI: 10.21099/TKBJM/1496164421
description: scientific article; zbMATH DE number 1880960