## Sturm-Liouville Equations with Singular Endpoints of Poincaré Rank Zero and One

##### Abstract

In this dissertation, my objective is to study, for x ∊ (0, ∞), the equations (1) (xy′)′ + (‐v²/x + λ x)y = 0, with a view to obtaining representations of solutions near the singular endpoints, connection formulas for solutions defined near one singular endpoint in terms of solutions defined at the other singular endpoint, and to make use of such information to prepare the way for spectral theoretic investigations involving the Titchmarsh-Weyl m ‐ function and spectral density function for equation (2). The work is motivated in part by the formulation of the “Sturm-Liouville Connection Problem” recently formulated by Charles Fulton in [47] (See p. 158.). In all of the differential equations investigated in this dissertation, it has been the classification of the singular endpoints that has played the central role in the methods that have been developed. The basic definition of Poincaré rank is as follows: For the 2nd order equation w" (x) + f(x)w′(x) + g(x)w(x) = 0, x ∊ (0, ∞), where f(x) and g(x) are rational functions for which either f(x) or g(x) or both f(x) and g(x) have poles at
x = x₀ ∊ [0, ∞). Then the singular point x = x₀ has Poincaré rank ℓ‐1 where ℓ is the least integer for which both (x‐ x₀)ʵ f(x) and (x‐x₀)²ʵ g(x) contain no powers of x ‐ x₀ in their denominators [83, p.148], [10]-[15]. To determine the Poincaré rank of a singularity at x = ∞, one may transfer this singularity to t = 0 by means of the inversion substitution x = 1/t and apply the scheme described above for the resulting equation. The rank of t = 0 is equal to the rank of x = ∞. A rank of 0 is the classical case of a regular singular point, and a rank of 1 is the case of an irregular singular point. Below is a summary of the content contained within each chapter of this dissertation. Chapter 1 addresses equations with regular singular endpoints whereby a new matrix method of solution is given with an application shown towards the Bessel equation of non-integral order. In Chapter 2 we investigate equation (2), which has irregular singular points of rank 1 [21] at both x =0 and x = ∞.
Here a new method is given to formulate principal and canonical non-principal solutions near the LP irregular singular endpoint x = 0 which are entire in λ. A similar approach is taken to make definitions of solutions near x = ∞ for Im λ > 0. Existence and uniqueness of the solutions is established. The Sturm-Liouville Connection problem is solved using the solutions defined at x = 0 and x = ∞. Chapter 3 addresses a new characterization of the spectral density function f(λ) given by Fulton, Pearson, and Pruess [45]. This method makes use of the Appell System, a companion linear system of ordinary differential equations. Here the spectral density function for a case involving the Bessel equation is obtained demonstrating the first nontrivial example of a spectral density function calculation using this new technique.