Eigenfunction Expansions for a Singular Eigenvalue Problem with Eigenparameter in the Boundary Condition

We study the singular boundary value problem $\begin{gathered} \tau (u) = \frac{1}{r}\left\{ { - \left( {pu'} \right)^\prime + qu} \right\} = \lambda u,\quad a \leqq x < b < \infty \hfill \\ - \beta _1 u(a) + \beta _2 \left( {pu'} \right)(a) = \lambda \left[ {\beta '_1 u(a) - \beta '_2 \left( {pu'} \right)(a)} \right] \hfill \\ \end{gathered} $ Conditions are placed on the coefficients which ensure the spectrum is bounded below and the essential spectrum is empty. An eigenf unction expansion theory is developed for a class $\mathcal{D}_1 $ of functions. For this class the convergence is uniform and absolute on compact intervals. When the eigenvalues are all nonnegative, the class $\mathcal{D}_1 $ is shown to be the domain of $A^{{1 / 2}} $, where A is the self -adjoint operator associated with the boundary value problem.