Eigenfunction expansion associated with the one-dimensional Schrödinger equation on semi-infinite time scale intervals

We prove the existence of a spectral function (spectral measure or orthogonality measure) for the one-dimensional Schrödinger equation on semi-infinite time scale intervals. A Parseval equality and an expansion formula in eigenfunctions are established in terms of the spectral function. The result obtained unifies and extends the well-known results on the existence of a spectral measure for the one-dimensional Schrödinger operator on the semi-axis and for semi-infinite Jacobi matrices.