Continuous spectrum eigenfunction expansions associated with the Rayleigh equation (Q1326938)

This paper considers the Rayleigh equation \((u - c) (\psi'' - \alpha^ 2 \psi) - u'' \psi = 0\), on the interval \(z \in (a,b)\), \(\psi (a) = \psi (b) = 0\), \(a,b\) are bounded, the function \(u = u(z)\) is given. Statements of the following results are given. Firstly, for each \(c \in (u(a), u(b))\) there exits a generalised eigenfunction \(\psi_ c\). Secondly, provided \(u(z)\) is somehow close to a linear function, there exists a bounded operator \(W\) with bounded inverse in \(L^ 2 (a,b)\) such that each function \(f \in C^{2 + \varepsilon} [a,b]\), with \(f(a) = f(b)\) satisfies the integral equation \(f + \int^ b_ a [W(f'' - \alpha^ 2f)] (z) \psi_{u(z)} du(z)\). Finally, a characterisation is given of the continuous spectrum eigenfunctions.