Cheap Sturm-Liouville Spectral Functions

软件包SLEDGE （Sturm-Liouville estimations Determined by Global errors -control）是目前唯一可以计算具有简单连续谱的奇异问题的奇异谱函数的近似的Sturm-Liouville代码。本研究采用了一种新的谱函数实变量公式，大大减少了计算量，从而节省了计算时间。新公式只需要计算展开式中一个解的零点，以及它在这些零点处的准导数。所产生的新代码将能够处理以下两种具有简单谱的奇异Sturm-Liouville问题：(1)具有有限截止值的左端点正则和右端点OSC-NONOSC；(2)具有有限截止值的NONOSC型左端点（极限圆或极限点）和正则奇异点和右端点OSC-NONOSC。在这个项目下，Sturm-Liouville问题软件的持续开发将为具有连续谱的Sturm-Liouville问题的谱密度函数提供更快速的例程。这样一种快速计算能力的存在必然会刺激和帮助在出现这类问题的广泛学科中的研究：(1)上述类型的问题经常出现在薛定谔波动力学中，物理学家发展了这一力学来研究原子和原子粒子的性质。(2)同样，在研究更复杂的分子时，量子化学家利用量子力学理论来描述分子中能量交换的性质。在海洋动力学领域，海洋声波理论引起了Sturm-Liouville问题的一些非常有趣的应用。(4)某些类型的喷水喷嘴涉及高速射流的流体流动问题，在有限范围内涉及上述类型的Sturm-Liouville问题。除了在数学以外的学科中广泛的应用之外，目前的研究也可以预期会刺激从事各种相关主题的纯数学家和应用数学家以及理论物理学家的活动，例如“逆谱理论”（试图从散射数据中重建势能函数），量子理论中的“共振问题”（识别和模拟正能量束缚态的问题），以及晶体理论中出现的“周期电位和带谱”。在中小型计算机上快速运行谱函数计算的能力将极大地帮助对Sturm-Liouville问题及其应用的理论和计算方面感兴趣的数学家、物理学家和计算机科学家的教学和培训。

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.