Eigenfunction expansions associated with pairs of ordinary differential expressions
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 found in: \textit{E. A. Coddington}, \textit{A. Dijksma} and \textit{H. S. V. de Snoo}, Proc. R. Soc. Edingb. Sect A 97. 59-71 (1984)]. The integral transform V:\({\mathcal H}\to L^ 2(\rho)\) associated with F differs from the usual generalized Fourier transform in that it is no longer an isometry but, in general, a contraction. In fact, \(F(\infty)=\lim_{\lambda \to \infty}F(\lambda)\) induces a splitting of \({\mathcal H}:\) \({\mathcal H}={\mathcal H}_ 0\oplus {\mathcal H}_ 1\oplus {\mathcal H}_ r,\) \({\mathcal H}_ 0=nullspace\) of F(\(\infty)\), \({\mathcal H}_ 1=nullspace\) of I-F(\(\infty)\) and \({\mathcal H}_ r={\mathcal H}\theta ({\mathcal H}_ 0\oplus {\mathcal H}_ 1),\) and we have \(\| Vf\|_{\rho}\leq \| f\|,\) \(f\in {\mathcal H}\), \({\mathcal H}_ 0=\{f\in {\mathcal H}| Vf=0\}\) and \({\mathcal H}_ 1=\{f\in {\mathcal H}| \| Vf\|_{\rho}=\| f\| \}.\) There exist examples where \({\mathcal H}_ r\neq \{0\}\). The paper also deals with the surjectivity of V. The results of this paper have been derived using a more abstract setting in some recent papers by \textit{H. Langer} and \textit{B. Textorius}: Spectral functions of a symmetric linear relation with a directing mapping, I, Proc. R. Soc. Edinb. Sect A 97, 165-176 (1984), and II, to appear.
generalized spectral family of orthogonal projections
scientific article; zbMATH DE number 3877832