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

Spectral Theory of Singular Hahn Difference Equation of the Sturm-Liouville Type

“…Usually, if we want to solve a partial differential equation using the Fourier method (i.e., the separation of variables) then we consider the problem of expanding an arbitrary function as a series of eigenfunctions. Hence the eigenfunction expanding problem has been studied extensively in the literature (see [2], [3], [4], [5], [6], [7], [13], [15], [16], [21], [22], [29], [30], [31], [34], [35], [36], [38], [39]).…”

Sturm–Liouville operators on time scales

“…The associated eigenfunction expansion was considered in [11], [16], [20] for the case of bounded time scales and for semi-unbounded time scales in [17], [18]. Here we will obtain it via classical Weyl-Titchmarsh theory thereby generalizing the presently best result from [18] where the case r = 1, p differentiable, and q continuous is treated. For further generalizations of Weyl-Titchmarsh theory to time scale systems see [25], [24] and the references therein.…”

“…[5], [6] and the references therein). In particular, Sturm-Liouville equations on time scales have attracted substantial interest (see, e.g., [1,2,3,9,10,11,14,15,16,17,18,19,20,21,23,28] and the references therein) since it contains both continuous Sturm-Liouville equations as well as their discrete analog, Jacobi equations, as special cases. However, efforts to unify these two cases go at least back to the seminal work of Atkinson [4] or the book by Mingarelli [22].…”