Topics in Spectral Theory and Nonlinear Equations

谱理论与非线性方程主题。mikhail shubin本项目将研究n维、子域和有边界的黎曼流形的拉普拉斯算子和薛定谔算子的谱理论，以及具有Dirichlet边界条件的黎曼流形。我们计划获得具有一般标量和矢量势的磁性薛定谔算子的谱离散性和严格正性的新的精确判据。我们将使用规范优化将许多问题简化为具有势的薛定谔算子族的问题。我们希望对（磁性）薛定谔算子的谱底和基本谱建立新的更精确的双边估计。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。特别地，我们将利用一维薛定谔算子的特征函数所满足的一阶演化PDE来研究一类函数中mKdV的解的构造，这些一维薛定谔算子的时变势满足KdV。我们还提出研究Arnold问题，即求有限群G的固定不可约表示对应于椭圆自伴随算子a的Dirichlet特征值的分布函数的渐近增长率，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估计。长期以来，薛定谔算符的谱一直被解释为量子粒子在与算符系数相关的电场和磁场中的能级。所提出的研究结果可以从量子粒子在不同能级上的局域化和稳定性来解释。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）方程和改进的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，我们期望用谱方法得到新的解类。

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.