Eigenfunction expansions associated with the one-dimensional schrödinger operator

We consider the form of eigenfunction expansions associated with the time-independent Schrödinger operator on the line, under the assumption that the limit point case holds at both of the infinite endpoints. It is well known that in this situation the multiplicity of the operator may be one or two, depending on properties of the potential function. Moreover, for values of the spectral parameter in the upper half complex plane, there exist Weyl solutions associated with the restrictions of the operator to the negative and positive half-lines respectively, together with corresponding Titchmarsh-Weyl functions. In this paper, we establish some alternative forms of the eigenfunction expansion which exhibit the underlying structure of the spectrum and the asymptotic behaviour of the corresponding eigenfunctions. We focus in particular on cases where some or all of the spectrum is simple and absolutely continuous. It will be shown that in this situation, the form of the relevant part of the expansion is similar to that of the singular half-line case, in which the origin is a regular endpoint and the limit point case holds at infinity. Our results demonstrate the key role of real solutions of the differential equation which are pointwise limits of the Weyl solutions on one of the half-lines, while all solutions are of comparable asymptotic size at infinity on the other half-line.