A fundamental approach to the generalized eigenvalue problem for self-adjoint operators

The generalized eigenvalue problem for an arbitrary self-adjoint operator is solved in a Gelfand triple consisting of three Hilbert spaces. The proof is based on a measure theoretical version of the Sobolev lemma, and the multiplicity theory for self-adjoint operators. As an application necessary and sufficient conditions are mentioned such that a self-adjoint operator in L₂(R) has (generalized) eigenfunctions which are tempered distributions.