Periodic ``weighted'' operators
type: Item
instance of: scholarly article
title: Periodic ``weighted'' operators
zbMATH Open document ID: 1029.34012
DOI: 10.1016/S0022-0396(02)00154-7
published in: Journal of Differential Equations
publication date: 2003-05-25T00:00:00Z
review text: The author and his collaborators, specifically M. Krein and P. Kargaev, extensively studied problems leading to second-order differential equations with periodic coefficients. These problems arise in studies of propagation of waves in a periodic medium, mechanics of the hydrogen atom in a periodic magnetic field, or vibration of crystals with periodic dielectric coefficients. This study, initiated by Lyapunov, was continued in the 1950s by M. Krein, who extended Lyapunov results to the \(2\times 2\) matrix case. It considers equations of the type \[ -f''(t)=\lambda b^2(t) f(t),\quad\lambda\in \mathbb{C},\;t\in\mathbb{R},\tag{1} \] with \(b(t)\geq 0\) and \(1\)-periodic, and \(\int^1_0 b(t) dt= 1\). The investigation of the inverse problem leads to the study of the weighted operator given by \(T_b f= -b^{-2}f''(t)\) posed in the \(L^2\) setting. This was studied by Lyapunov and later by M. Krein. They found that the spectrum of \(T_b f\) is absolutely continuous and consists of intervals \(\sigma_n\), which are separated by gaps \(\gamma_n= (\lambda^-_n, \lambda^+_n)\). If the gap is degenerate, then these intervals merge. A Lyapunov function is defined in terms of the elementary solutions \(y_1\), \(y_2\) to system (1), with initial conditions imposed on these functions at \(t= 0\). The Lyapunov function is defined as \(D(\lambda)= {1\over 2}(y_2'(1, \lambda)+ y_1(1,\lambda))\). The author rewrites system (1) by introducing the variable \(x(t)= \int^t_0 b(\tau) d\tau\) and variable \(\rho(x)= b(t(x))^{{1\over 2}}\). The well-known substitution \(q(t)= \rho'/\rho\) completes this introduction of new variables. The corresponding operator \(T_\rho f= -\rho^{-2}(\rho^2 f')'= -f''- 2qf'\) is even, general with nonsmooth coefficients. The author considers the case when these coefficients are smooth, and derives asymptotic estimates on the lengths of the gaps in the spectrum. His proofs use the analysis of a conformal map called quasi-momentum. The Lyapunov function plays …
reviewed by: Vadim Komkov
Mathematics Subject Classification ID: 34A55
Mathematics Subject Classification ID: 78A25
Mathematics Subject Classification ID: 34L99
zbMATH DE Number: 1915522
zbMATH Keywords: inverse problems
zbMATH Keywords: second-order ordinary differential equations
zbMATH Keywords: periodic coefficients
zbMATH Keywords: waves in a periodic medium
zbMATH Keywords: mechanics of the hydrogen atom
zbMATH Keywords: periodic magnetic field
zbMATH Keywords: vibration of crystals
zbMATH Keywords: periodic dielectric coefficients
author: E. Korotyaev
MaRDI profile type: Publication
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full work available at URL: s0022-0396(02)00154-7
OpenAlex ID: W2040289542
description: scientific article; zbMATH DE number 1915522