Verification of Spectral Analysis of a Self-adjoint Differential Operator “Following Landau and Lifshitz” by Means of Its Green Function

We examine an example of a self-adjoint ordinary differential operator going back to Naimark. This operator is remarkable because its point and continuous spectra intersect. We find the spectrum and eigenfunctions of this operator “following Landau and Lifshitz,” i.e., following the rules stated in their book Quantum Mechanics and based on plausible heuristic physical arguments and analogies with linear algebra, which, to our knowledge, have not been rigorously mathematically justified so far. Then we adduce arguments in support of reasonableness of the results obtained by this method, which is conventional for physicists. The arguments are based on the analysis of the independently calculated Green function of the operator.