Topics in Spectral Theory and Nonlinear Equations

谱理论和非线性方程主题

• 批准号: 0600196
• 负责人: Mikhail Shubin
• 金额: --
• 依托单位: Northeastern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0600196&HistoricalAwards=false
• 关键词: Topics; Spectral; Theory; Nonlinear; Equations

Topics in Spectral Theory and Nonlinear Equations.Abstract of Proposed ResearchMikhail ShubinThis project will study topics in the spectral theory of the Laplace and Schroedinger operators in n-dimensions, on subdomains and on Riemannian manifolds with boundary, with Dirichlet boundary conditions. We plan to obtain new and precise criteria for discreteness of spectra and strict positivity for the magnetic Schroedinger operators, with general scalar and vector potentials. We will use a gauge optimization to reduce many issues to those for a family of the usual Schroedinger operators with a potential. We wish to establish new and more precise two-sided estimates for the bottom of the spectrum and essential spectrum of (magnetic) Schroedinger operators. We plan to study applications of the spectral theory of one-dimensional Schroedinger operators to KdV and mKdV equations. In particular the construction of solutions of mKdV in classes of functions which may grow at infinity with respect to the space variable will be investigated using a first order evolution PDE satisfied by eigenfunctions of the one-dimensional Schroedinger operators whose time-dependent potentials satisfy KdV. We also propose to investigate Arnold's problem of finding the asymptotic growth rate of the distribution function of the Dirichlet eigenvalues corresponding to a fixed irreducible representation of a finite group G for an elliptic self-adjoint operator A on a compact manifold with boundary and with a G-action commuting with A. This will be done using the method of approximate spectral projection, which should also provide an estimate of remainder in the asymptotic expansion.The spectrum of a Schroedinger operator has long been interpreted in terms of the energy levels of a quantum particle in the electric and magnetic fields associated with the coefficients of the operator. The results of the proposed research may be interpreted in terms of the localization and stability of the quantum particle at various energy levels. We plan to study influence of the symmetries of different states on the asymptotic distribution of the corresponding energy levels. The Korteweg - de Vries (KdV) and modified Korteweg - de Vries (mKdV) equations are important model equations of various phenomena in non-linear dynamics and we expect to obtain new classes of solutions by using spectral methods.

谱理论和非线性方程主题。拟议研究摘要Mikhail Shubin 该项目将研究 n 维、子域和带边界的黎曼流形以及狄利克雷边界条件的拉普拉斯和薛定谔算子的谱理论主题。我们计划为具有一般标量和矢量势的磁薛定谔算子获得光谱离散性和严格正性的新的精确标准。 我们将使用规范优化来减少对具有潜力的普通薛定谔算子系列的许多问题。 我们希望对（磁）薛定谔算子的谱底和基本谱建立新的、更精确的两侧估计。我们计划研究一维薛定谔算子的谱理论在 KdV 和 mKdV 方程中的应用。 特别是，将使用一维薛定谔算子的本征函数所满足的一阶演化偏微分方程来研究相对于空间变量可能无限增长的函数类中 mKdV 解的构造，该算子的瞬态势满足 KdV。我们还建议研究阿诺德问题，即找到与有限群 G 的固定不可约表示相对应的狄利克雷特征值分布函数的渐近增长率，该椭圆自伴算子 A 在具有边界的紧流形上并且具有与 A 交换的 G 动作。这将使用近似谱投影的方法来完成，该方法还应提供渐近展开中的余数的估计。 薛定谔算子的谱长期以来一直根据与算子系数相关的电场和磁场中量子粒子的能级来解释。 所提出的研究结果可以用量子粒子在不同能级的局域化和稳定性来解释。 我们计划研究不同态的对称性对相应能级渐进分布的影响。 Korteweg - de Vries (KdV) 和修正的 Korteweg - de Vries (mKdV) 方程是非线性动力学中各种现象的重要模型方程，我们期望通过使用谱方法获得新类别的解。