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Inverse problems in quantum mechanics.

机译：量子力学中的逆问题。

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摘要

Three topics are investigated in this thesis. (1) In the first part of the thesis, the inverse problem of extracting a quantum mechanical potential from laboratory data is studied from the perspective of determining the amount and type of data capable of giving a unique answer. Bound state spectral information and expectation values of time-independent operators are used as data. The Schroedinger equation is treated as finite dimensional and for these types of data there are algebraic equations relating the unknowns in the system to the experimental data. As these equations are polynomials in the unknown parameters of the system, it is possible to determine the multiplicity of the solution set. With a fixed number of unknowns the effect of increasing the number of equations on the multiplicity of solutions is assessed. We show that if one has more equations than the number of unknowns, generically a unique solution exists. Several examples illustrating these results are provided. (2) In the second part of the thesis we introduce a family of approximation methods: High dimensional model representations. A systematic mapping procedure between the inputs and outputs is prescribed to reveal the hierarchy of correlations amongst the input variables. It is argued that for most well defined physical systems, only relatively low order correlations of the input variables are expected to have an impact upon the output. The high dimensional model representations (HDMR) utilize this property to present an exact hierarchical representation of the physical system. Application of the HDMR tools can dramatically reduce the computational effort needed in representing the input-output relationships of a physical system. Selected applications of the HDMR concept are presented along with a discussion of its general utility. HDMRs can be classified as non-regressive, non-parametric learning networks. (3) The last part of the thesis is on a fast algorithm for the solution of a polynomial system of equations. Its relation to Buchberger's algorithm of finding the variety defined by a system of polynomials over the field of real numbers is discussed.
机译：本文研究了三个主题。 （1）在论文的第一部分，从确定能够给出唯一答案的数据的数量和类型的角度研究了从实验室数据中提取量子力学势的反问题。绑定状态频谱信息和与时间无关的算符的期望值用作数据。 Schroedinger方程被视为有限维，对于这些类型的数据，存在代数方程，将系统中的未知数与实验数据联系起来。由于这些方程是系统未知参数中的多项式，因此可以确定解集的多重性。在未知数目固定的情况下，评估了增加方程组数量对解的多重性的影响。我们证明，如果一个方程具有比未知数更多的方程，则通常存在唯一解。提供了说明这些结果的几个示例。 （2）在论文的第二部分，我们介绍了一系列近似方法：高维模型表示。规定了输入和输出之间的系统映射过程，以揭示输入变量之间的相关性层次。有人认为，对于大多数定义良好的物理系统，只有输入变量的较低阶相关性才有望对输出产生影响。高维模型表示（HDMR）利用此属性来呈现物理系统的精确层次表示。 HDMR工具的应用可以大大减少表示物理系统的输入输出关系所需的计算量。介绍了HDMR概念的选定应用，并讨论了其一般用途。 HDMR可以分类为非回归，非参数学习网络。 （3）论文的最后一部分是关于求解多项式方程组的快速算法。讨论了它与Buchberger算法的关系，该算法用于查找由实数域上的多项式系统定义的变体。

著录项

  • 作者

    Alis, Omer Faruk.;

  • 作者单位

    Princeton University.;

  • 授予单位 Princeton University.;
  • 学科 Mathematics.
  • 学位 Ph.D.
  • 年度 2001
  • 页码 125 p.
  • 总页数 125
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 数学;
  • 关键词

  • 入库时间 2022-08-17 11:47:19

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