Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations
type: Item
instance of: scholarly article
title: Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations
zbMATH Open document ID: 0928.35161
DOI: 10.2307/121001
published in: Annals of Mathematics. Second Series
publication date: 1999-06-20T00:00:00Z
full work available at URL: 148 2.html
review text: The author's main problem is determination and characterization of solutions of integrable, but primarily linear, perturbed Hamiltonian equations. Specifically the author is interested in extending the classical KAM theory of existence of invariant tori. In this presentation the finite dimensional tori ``live'' in infinite-dimensional space. This may lead to small divisor computational instability problems. In one spatial dimension the natural frequencies are well-separated from each other. But in dimensions greater than one phenomenon of clustering of natural frequencies occurs, generally making the techniques used in derivation of KAM inapplicable. In fact the natural frequencies may have unbounded multiplicities. The author comments that at the present time not much is known about solutions of 3- or higher-dimensional nonlinear Schrödinger models. The case of known periodic solutions that is of persistence of \(n\)-dimensional tori is much easier. For example consider the equation: \(iu_t+Au+ \varepsilon\partial H/\partial\overline u=0\), where \(A\) is a selfadjoint operator, \(H(u, \widetilde u)\) is either a polynomial or real-analytic, and \(A=-\Delta+M_\alpha\) is the Laplacian plus a Fourier multiplier vector: \(M_\sigma\). Eigenfunction basis are exponentials \(e^{inx}\), and eigenvalues are \(\lambda_n =n^2+ \sigma_n\) with \(\lim_{n\to \infty}(\sigma_n) =0\). The author compares the Sturm-Liouville operator with \(A=d^2/dx^2 +V(x)\), where one encounters localization property of the eigenfunctions. This argument fails in dimensions higher than one. However in the special case when \(V(x)=V_1 (x_1)+V_2(x_2)+ \dots V_d(x_d)\) the solutions of unperturbed linear equation correspond to a flow on a \(b\)-dimensional torus. The aim of this paper is to generalize persistent properties of these tori in phase space. The author reduces weight functions, which allow some simplifications of this problem, and studies resulting dependency of the new eigenvalues on Fou…
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 81Q15
zbMATH DE Number: 1267995
zbMATH Keywords: two-dimensional KAM theory
zbMATH Keywords: Schrödinger equation
zbMATH Keywords: Hamiltonian systems
author: Jean Bourgain
reviewed by: Vadim Komkov
MaRDI profile type: Publication
OpenAlex ID: W1967673736
Wikidata QID: Q97672473
description: scientific article; zbMATH DE number 1267995