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

Highlights: → We prove existence of a spectral measure for second-order delta dynamic equations on time scales. → The result implies, in particular, existence of an orthogonality measure in the discrete case. → An expansion in eigenfunctions formula is established in terms of the spectral measure. → The result extends the well-known results to the more general context of time scales. - Abstract: 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.