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.

频谱理论和非线性方程的主题。拟议的研究米哈伊尔·舒本（Shubinthis）项目将研究拉普拉斯（Laplace）和施罗丁格（Schroedinger）操作员在n级，子域和带有边界的riemannian歧管的光谱理论中的主题。我们计划获得具有一般标量和向量电位的磁性schroedinger操作员的光谱离散性和严格阳性的新标准。 我们将使用量规优化来减少对通常具有潜力的施罗辛格运营商家庭的许多问题。 我们希望为（磁性）Schroedinger操作员的底部和基本频谱建立新的，更精确的双向估计。我们计划研究一维Schroedinger操作员在KDV和MKDV方程的光谱理论的应用。 特别是，将使用一维Schroedinger运算符的特征函数满足其时间依赖性势能满足KDV的一维schroedinger操作员满足的一阶进化PDE来研究MKDV解决方案在无穷大的函数类别中的构建。 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长期以来，用与操作员系数相关的电场和磁场中的量子粒子的能级来解释了施罗辛格操作员的光谱。 拟议研究的结果可以根据各种能级的量子粒子的定位和稳定性来解释。 我们计划研究不同状态的对称性对相应能级渐近分布的影响。 Korteweg -de Vries（KDV）和改良的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，我们希望使用光谱方法获得新的解决方案。