Eigenfunction expansions associated with pairs of ordinary differential expressions
type: Item
instance of: scholarly article
title: Eigenfunction expansions associated with pairs of ordinary differential expressions
zbMATH Open document ID: 0551.47008
DOI: 10.1016/0022-0396(85)90119-6
author: A. Dijksma
author: H. S. V. de Snoo
published in: Journal of Differential Equations
publication date: 1985-01-01T00:00:00Z
review text: Let L and M be two formally symmetric \(m\times m\) systems of ordinary differential expressions of orders n and \(\nu\) respectively, defined on an open interval \(\iota\) \(\subset {\mathbb{R}}\) with coefficients which are sufficiently often differentiable and such that the leading coefficient of M is invertible on \(\iota\) and that, at least in the case \(n>\nu\), the same is true for the leading coefficient of L. In the paper we deduce - under the assumption that M is in some sense locally positive on \(\iota\)- expansions of functions in terms of a fundamental system of solutions s(\(\ell)\) of the equations: \((L-\ell M)s(\ell)=0\) if \(n>\nu\) (right definite case) and \((M-\ell L)s(\ell)=0\) if \(n<\nu\) (left definite case). The case in which L and M are also assumed to be regular has been worked out in detail by \textit{E. A. Coddington} and \textit{H. S. V. de Snoo}, Regular boundary value problems associated with pairs of ordinary differential expressions, Lecture Notes Math. 858 (1981; Zbl 0464.34003). The method we use for the general case is based on the paper by \textit{E. A. Coddington} and \textit{H. S. V. de Snoo}, J. Differ. Equations 35, 129- 182 (1980; Zbl 0398.34022), and is as follows: From the positivity condition on M it follows that there are Hilbert spaces \({\mathcal H}={\mathcal H}_ H\) associated with selfadjoint realizations H of M in \(L^ 2(\iota)\) and inner product generated by \([f,g]=\int_{\iota}g^*(Hf),\) f, \(g\in D(H)\), the domain of H. In such a space \({\mathcal H}\) we consider the minimal symmetric operator S which basically looks like \(H^{-1}L\) and an arbitrary generalized spectral family of orthogonal projections \(F=(F(\lambda))_{\lambda \in {\mathbb{R}}}\) associated with S. From the fact that F can be expressed in terms of s(\(\ell)\) and a spectral matrix \(\rho\) on \({\mathbb{R}}\) we obtain eigenfunction expansion theorems for all elements in \({\mathcal H}\). [Pointwise convergence of these expansions can be…
Mathematics Subject Classification ID: 47A70
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 34G10
zbMATH DE Number: 3877832
zbMATH Keywords: minimal symmetric operator
zbMATH Keywords: generalized spectral family of orthogonal projections
zbMATH Keywords: spectral matrix
zbMATH Keywords: eigenfunction expansion
zbMATH Keywords: integral transform
MaRDI profile type: Publication
full work available at URL: 0022-0396(85)90119-6
OpenAlex ID: W2131482820
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description: scientific article; zbMATH DE number 3877832