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Differential operators with spectral parameter incompletely in the boundary conditions. (English) Zbl 0715.34133
This paper deals with the linear Hamiltonian system \((1)\quad Jy'=(\lambda A(x)+B(x))y,\) \(-\infty <a\leq x<b\leq \infty\), \(y=(y_ 1,y_ 2)^ T\), together with boundary conditions \((2)\quad \alpha_ 1y_ 1(a)-\alpha_ 2y_ 2(a)=\lambda (\alpha '_ 1y_ 1(a)-\alpha '_ 2y_ 2(a)).\) In (1)-(2), \(J=\left( \begin{matrix} 0\\ I\end{matrix} \begin{matrix} -I\\ 0\end{matrix} \right)\) where I is the \(n\times n\) identity, A(x) and B(x) are 2n\(\times 2n\) symmetric matrices with A(x)\(\geq 0\), the \(y_ k\) are n-vectors, and the \(\alpha_ k\) and \(\alpha_ k'\) are \(n\times n\) matrices. It is assumed that the number r of linearly independent boundary conditions in (2) which involve \(\lambda\) is less than or equal to n. Numerous problems in physics and engineering are of the form (1)- (2), and some of these are discussed in this paper. Conditions are placed on the \(\alpha_ k\) and \(\alpha_ k'\) which give (1)-(2) a self-adjoint formulation in a Hilbert space. The Green’s function and Titchmarsh-Weyl coefficient M(\(\lambda\)) are constructed both for regular and singular problems. Finally the nature of the spectrum of (1)-(2) is connected with the singular structure of M(\(\lambda\)) by proving a Chaudhuri-Everitt type theorem [J. Chaudhury and W. N. Everitt, Proc. Roy. Soc. Edinb., Sect. A 68 (1967-68), 95-119 (1969; Zbl 0194.171)].
Reviewer: D.B.Hinton

MSC:
34L05 General spectral theory of ordinary differential operators
34B20 Weyl theory and its generalizations for ordinary differential equations
34B27 Green’s functions for ordinary differential equations
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