Cheap Sturm-Liouville Spectral Functions

Florida Institute Of Technology, Melbourne FL

Investigators

Abstract

The software package SLEDGE (Sturm-Liouville Estimates Determined by Global Error-control) is currently the only Sturm- Liouville code that can compute approximations to the singular spectral function of singular problems having a simple continuous spectrum. The present research will greatly reduce the amount of computation required, and thereby the computer time, by making use of a new real-variable formula for the spectral function. The new formula requires only that the zeros of the one solution that comes into the expansion formula, and its quasi-derivative at these zeros, be computed. The new code produced will be able to handle the following two cases of singular Sturm-Liouville problems having simple spectrum: (1) left endpoint regular and right endpoint OSC-NONOSC with finite cut-off value and (2) left endpoint of NONOSC type (Limit Circle or Limit Point) and a Regular Singular Point and right endpoint OSC-NONOSC with finite cut-off value. The continuing development of software for Sturm-Liouville problems to be done under this project will yield much more rapid routines for the spectral density functions for Sturm-Liouville problems that have a continuous spectrum. The existence of such a rapid computational capability is bound to stimulate activity and aid research in a wide range of disciplines where such problems arise: (1) Problems of the above type frequently arise in the Schroedinger wave mechanics which physicists developed to study the nature of atoms and atomic particles. (2) Similarly, in the study of more complicated molecules quantum chemists make use of quantum mechanical theory to describe the nature of energy exchange in molecules. (3) In the area of ocean dynamics the theory of acoustic waves in the ocean gives rise for some very interesting applications of Sturm-Liouville problems. (4) Certain types of nozzles for spraying water involve fluid flow problems for high speed jets from the nozzle which involve the above type of Sturm-Liouville problem over a finite range. In addition to a wide variety of applications in disciplines outside of mathematics, the current research can also be expected to stimulate the activity of pure and applied mathematicians and theoretical physicists who work on various related topics such as "Inverse Spectral Theory" (the attempt to reconstruct potential functions from scattering data), "Resonance Problems" in quantum theory (the problems of identifying and modelling positive energy bound states), and "Periodic Potentials and Band Spectra" which arise in the theory of crystals. The ability to run spectral function calculations quickly on small to medium size computers will greatly aid in the teaching and training of mathematicians, physicists and computer scientists interested in the theoretical and computational aspects of Sturm-Liouville problems and their applications.