Fundamental solutions and eigenvalues of a differential operator with indefinite weight (Q2910707)

The author studies the fourth-order eigenvalue problem NEWLINE\[NEWLINE(a(x) u''(x))'' + q(x) u(x) = \lambda r(x) u(x)\quad \text{on } [-1,0) \cup (0,1]NEWLINE\]NEWLINE equipped with \(\lambda\)-dependent boundary conditions at \(-1\) and \(1\) and with so-called transmission conditions at \(0\). Here, \(q, r \in L^1[-1,1]\) are real functions and \(a\) has the particular form \(a(x) = a_1^4\) for \(x \in [-1,0)\) and \(a(x) = a_2^4\) for \(x \in (0,1]\) with \(a_1, a_2 > 0\). A realization of the problem by means of a self-adjoint linear operator \(A\) in a Krein space \(K\) is presented. Although the notation remains a little bit uncertain, \(K\) seems to be a weighted \(L^2\)-space essentially determined by the indefinite weight function \(r\) plus a weighted four-dimensional space. A set of four functions is constructed forming a fundamental system of solutions such that the corresponding Wronskian \(W(\lambda)\) is independent of \(x\) and its zeroes coincide with the eigenvalues of the original problem.