On the asymptotics of solutions to the second initial boundary value problem for Schrödinger systems in domains with conical points.
type: Item
instance of: scholarly article
title: On the asymptotics of solutions to the second initial boundary value problem for Schrödinger systems in domains with conical points.
zbMATH Open document ID: 1274.35023
author: N. M. Hung
author: Hoang Viet Long
author: Nguyen Thi Kim Son
published in: Applications of Mathematics
publication date: 2013-05-13T00:00:00Z
full work available at URL: http://hdl.handle.net/10338.dmlcz/143135
review text: The paper deals with the second initial boundary value problem for a general Schrödinger system of order \(2m\) in \(\Omega \times (0,\infty )\) where the boundary \(\partial \Omega \) has some conical point, say \(x=0\). While existence and uniqueness of weak solutions was proved in a previous paper (see [\textit{N. M. Hung} and \textit{N. T. K. Son}, Electron. J. Differ. Equ. 2008, Paper No. 35, 11 p. (2008; Zbl 1170.35360)]) the authors now study the asymptotic behavior of generalized solutions in a neighbourhood of the conical point. To this end they derive an elliptic eigenvalue problem depending on the parameter \(t\) which is defined on a part \(G\) of the sphere \(S^{n-1}\) generated by the cone. If this eigenvalue problem in a given strip has semisimple eigenvalues with invariant multiplicity for all \(t\in [0,\infty ]\) there is proven a representation of a weak solution in the cone in terms of powers of \(r=| x| \) and vector polynomials of \(\ln r\). The proof of this result is the main part of the paper. In the final section, as an example the Neumann problem for the classical Schrödinger equation in quantum mechanics is discussed for the cases of space dimensions \(n=2\), \(n=3\) and \(n\geq 4\), respectively.
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35G99
Mathematics Subject Classification ID: 35C20
Mathematics Subject Classification ID: 35Q40
zbMATH DE Number: 6162813
zbMATH Keywords: second initial boundary value problem
zbMATH Keywords: Schrödinger system
zbMATH Keywords: generalized solution
zbMATH Keywords: regularity
zbMATH Keywords: asymptotic behavior
MaRDI profile type: Publication
OpenAlex ID: W3162556322
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DOI: 10.1007/S10492-013-0003-9
description: scientific article; zbMATH DE number 6162813