Zur Vollständigkeit des Systems der Eigenfunktionen irregulärer Eigenwertprobleme mit \(\lambda\)-abhängigen Randbedingungen
type: Item
instance of: scholarly article
title: Zur Vollständigkeit des Systems der Eigenfunktionen irregulärer Eigenwertprobleme mit \(\lambda\)-abhängigen Randbedingungen
zbMATH Open document ID: 0536.34016
DOI: 10.4171/ZAA/106
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1984-01-01T00:00:00Z
review text: We consider irregular eigenvalue problems \[ (1)\quad y^{(n)}+\sum^{n}_{\nu =2}p_{\nu}(x)y^{(n-\nu)}=\lambda y, \] \[ (2)\quad \sum^{n-1}_{j=0}\alpha_{\nu j}(\lambda)y^{(j)}(0)=0,\quad 1\leq \nu \leq m>n-m, \] \[ \sum^{n- 1}_{j=1}[\alpha_{\nu j}(\lambda)y^{(j)}(0)+\beta_{\nu j}(\lambda)y^{(j)}(1)]=0,\quad m+1\leq \nu \leq n, \] with polynomials \(\alpha_{\nu j}\) and \(\beta_{\nu j}\) and summable functions \(p_{\nu}\). Eigenvalue problems of this type are irregular - that means that the Green's function of these problems is growing exponentially for \(| \lambda | \to \infty.\) Assuming that the boundary conditions (2) have different weights, we give asymptotic estimates for the eigenvalues of problem (1),(2), and we prove that the system of eigenfunctions and associated functions of (1), (2) is complete in \(L_ 2[0,1]\).
Mathematics Subject Classification ID: 34L99
zbMATH DE Number: 3851525
zbMATH Keywords: irregular eigenvalue problems
zbMATH Keywords: completeness of eigenfunctions
zbMATH Keywords: Green's function
zbMATH Keywords: weights
author: Gerhard Freiling
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/zaa/106
OpenAlex ID: W2565663766
description: scientific article; zbMATH DE number 3851525