Existence of a spectral measure for second-order delta dynamic equations on semi-infinite time scale intervals

In this study, we prove existence of a spectral measure (or orthogonality measure) for second-order delta dynamic equations on semi-infinite time scale intervals. A Parseval equality and an expansion in eigenfunctions formula are established in terms of the spectral measure. The result obtained unifies the well-known results on existence of a spectral measure for Sturm-Liouville operators on the real semi-axis and for semi-infinite Jacobi matrices, and extends them to variety of numerous time scales which may, in particular, be fractals.