Cheap Sturm-Liouville Spectral Functions

廉价的 Sturm-Liouville 谱函数

• 批准号: 0109022
• 负责人: Charles Fulton
• 金额: $ 8.5万
• 依托单位: Florida Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-15 至 2006-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0109022&HistoricalAwards=false
• 关键词: Cheap; Sturm; Liouville; Spectral; Functions

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.

软件包SLEDGE（Sturm-Liouville Estimates Determined by Global Error-Control）是目前唯一的Sturm-Liouville码，其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。本研究将大大减少所需的计算量，从而减少计算机的时间，利用一个新的实变量公式的谱函数。新公式只需要计算展开公式中的一个解的零点及其在这些零点处的准导数。所产生的新代码将能够处理具有简单谱的奇异Sturm-Liouville问题的以下两种情况：（1）具有有限临界值的左端点常规和右端点OSC-NOOSC，以及（2）NOOSC型左端点（极限圆或极限点）和一个规则奇点和具有有限截止值的右端点OSC-NOOSC。刘维问题将在这个项目下完成将产生更快速的例程的频谱密度函数的Sturm-Liouville问题，有一个连续的频谱。这种快速计算能力的存在必然会刺激活动，并在出现此类问题的广泛学科中帮助研究：（1）上述类型的问题经常出现在物理学家为研究原子和原子粒子的性质而开发的薛定谔波动力学中。(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。(3)在海洋动力学领域，海洋中的声波理论引起了Sturm-Liouville问题的一些非常有趣的应用。(4)用于喷水的某些类型的喷嘴涉及来自喷嘴的高速射流的流体流动问题，其涉及有限范围内的上述类型的Sturm-Liouville问题。除了在数学以外的学科中的各种应用外，目前的研究还有望刺激从事各种相关主题（如“逆谱理论”）的纯数学家和应用数学家以及理论物理学家的活动。（试图从散射数据重建势函数），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“周期势和能带谱”。在中小型计算机上快速运行谱函数计算的能力将极大地有助于对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。