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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