( 18 ) The eigenvalue problem for ordinary differential equations of the second order and Heisenberg's theory of S-matrices

Recently E. C. Titchmarsh has treated the theory of expansion of arbitrary functions in terms of the eigenfunctions of a differential operator of the second order by a new method and obtained results of importance for applications.' The method of Titchmarsh is based solely on the calculus of residues. In the present paper we shall first give another proof of Titchmarsh's results based, on the general theory of linear operators in Hilbert space and, secondly, applying them to differential equations of Schrodinger type, show that a theorem of leisenberg2 concerning the S-matrix can be founded on these results.