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Green function of discontinuous boundary-value problem with transmission conditions. (English) Zbl 1146.34061

Let \((a,b)\) be a bounded interval, \(c\in (a,b)\) and let \(q\) be a real-valued function on \([a,b]\) which is continuous on \([a,c)\) and \((c,b]\). The authors study the Sturm-Liouville equation
\[ -u^{\prime\prime}+qu=\lambda u \]
subject to \(\lambda\)-linear boundary conditions at the endpoints \(a\) and \(b\), and, in addition, they impose a \(\lambda\)-linear interface condition at the point \(c\). Green’s function of such a discontinuous boundary value problem is expressed in terms of fundamental solutions on \((a,c)\) and \((c,b)\), respectively, and a formula for its asymptotic behaviour is given. The \(\lambda\)-linear boundary value problem leads in a standard way to a selfadjoint operator \(A\) in the Hilbert space \(L^2((a,b))\oplus\mathbb C^3\) such that the eigenvalues of \(A\) coincide with the eigenvalues of the problem under consideration. This linerization \(A\) is constructed explicitely and its resolvent is computed.

MSC:

34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
34B27 Green’s functions for ordinary differential equations
34B24 Sturm-Liouville theory
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