On a new method for solving irregular problems
type: Item
instance of: scholarly article
title: On a new method for solving irregular problems
zbMATH Open document ID: 0914.34073
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-11-29T00:00:00Z
review text: The author deals with the Regge problem \[ L(\lambda, D)u:= \lambda^2u(x)- u''(x)+ Q(x) u(x)= f(x),\quad x\in (0,1),\tag{1} \] \[ L_1(\lambda) u:=\lambda u(1)+ u'(1)= f_1,\quad L_2u:= u(0)= f_2,\tag{2} \] where \(f(x)\) and \(Q(x)\) are given complex-valued functions, \(f_1\), \(f_2\) are given complex numbers. This problem arises in scattering theory and it is irregular in the sense of Birkhoff. It is shown that (1), (2) is coercive inside the right half-plane and it is coercive with a defect inside the left half-plane. In the papers by \textit{T. Regge} [Nuovo Cimento, X. Ser. 8, 671-679 (1958; Zbl 0080.41903) and 9, 491-503 (1958; Zbl 0081.43002)] completeness of eigenfunctions and asymptotics of eigenvalues are established in the case of the real-valued potential \(Q(x)\). Here, the author suggests a new approach to the investigation of the problem (1), (2). He presents algebraic conditions for coercive solvability with a defect with respect to the spectral parameter. The twofold completeness of root functions of the Regge problem is proven.
Mathematics Subject Classification ID: 34L25
Mathematics Subject Classification ID: 34L10
Mathematics Subject Classification ID: 34B05
zbMATH DE Number: 1215720
zbMATH Keywords: spectral parameter
zbMATH Keywords: irregular
zbMATH Keywords: coercivity
zbMATH Keywords: Regge problem
zbMATH Keywords: scattering theory
zbMATH Keywords: completeness
zbMATH Keywords: eigenfunctions
zbMATH Keywords: asymptotics
zbMATH Keywords: eigenvalues
author: Sasun Yakubov
reviewed by: Irena Rachůnková
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5876
OpenAlex ID: W1991574455
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DOI: 10.1006/JMAA.1997.5876
description: scientific article; zbMATH DE number 1215720