This paper deals with the theory of point interactions for the one-dimensional Schrödinger equation. The familiar example of the δ-potential V(x) = gδ(x−x0), for which the transfer matrix across the singularity (point transfer matrix) is given by
is extended to cover cases in which the transfer matrix M(z) is dependent on the (complex) spectral parameter z, and which can be obtained as limits of transfer matrices across finite intervals for sequences of approximating potentials Vn.
The case of point transfer matrices polynomially dependent on z is treated in detail, with a complete characterization of such matrices and a proof of their factorization as products of point transfer matrices linearly dependent on z.
The theory presented here has applications to the study of point interactions in quantum mechanics, and provides new classes of point interactions which can be obtained as limiting cases of regular potentials.