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.


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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