Spectral parameter power series for Sturm-Liouville equations on time scales

Keywords: Sturm–Liouville Spectral parameter Taylor monomials Time scale a b s t r a c t We will derive formulas for finding two linearly independent solutions of the Sturm–Liouville dynamic equation. We will give several examples. In particular, the q-difference equation which has important applications in quantum theory will be presented. In this paper we obtain a spectral parameter power series representation for the solutions of the Sturm–Liouville dynamic equation ðpðtÞx D ðtÞÞ D þ qðtÞxðrðtÞÞ ¼ krðtÞxðrðtÞÞ ð1:1Þ in terms of a non-vanishing solution of the equation ðpðtÞx D ðtÞÞ D þ qðtÞxðrðtÞÞ ¼ 0; ð1:2Þ where p 2 CðT; KÞ and pðtÞ – 0 for all t 2 T; q; r 2 C rd ðT; KÞ (defined below), and k 2 K is a constant, where T denotes a time scale unbounded from above, i.e., sup T ¼ 1, and K is the set of complex numbers. Problems of this sort were considered for the differential equation ðpðtÞx 0 ðtÞÞ 0 þ qðtÞxðtÞ ¼ krðtÞxðtÞ and for the difference equation DðpðnÞDxðnÞÞ þ qðnÞxðn þ 1Þ ¼ krðnÞxðn þ 1Þ in several papers (see the very nice papers by Campos and Kravchenko [3] and Kravchenko and Porter [8]). Recall that a time scale T is an arbitrary nonempty closed subset of the real numbers R. The most well-known examples are T ¼ R; T ¼ Z, and T ¼ q Z :¼ fq n : n 2 Zg S f0g, where q > 1. The forward and backward jump operators are defined by respectively, where inf ; :¼ sup T and sup ; :¼ inf T. A point t 2 T is said to be left-dense if t > inf T and qðtÞ ¼ t, right-dense if t < sup T and rðtÞ ¼ t, left-scattered if qðtÞ < t, and right-scattered if rðtÞ > t. A function f that is defined on a time scale T is called rd-continuous if it is continuous at every right-dense point and if the left-sided limit exists (finite) at every 0096-3003/$-see front matter Ó 2012 Elsevier Inc. All rights reserved.