Topics in Spectral Theory and Nonlinear Equations

谱理论和非线性方程的主题。拟议的研究摘要米哈伊尔·舒宾这个项目将研究n维拉普拉斯和薛定谔算子的谱理论的主题，关于子域和带边界的黎曼流形，带Dirichlet边界条件。我们计划得到具有一般标量势和矢量势的磁性薛定谔算符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来将许多问题归结为一类具有潜力的常见薛定谔算符的问题。我们希望对(磁)薛定谔算子的谱底部和本质谱建立新的、更精确的双边估计。我们计划研究一维薛定谔算子谱理论在KdV和mKdV方程中的应用。特别地，我们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mKdV解的构造，这些函数类可以关于空间变量无限增长。我们还建议研究Arnold问题，即求出紧致流形上与A交换的椭圆自伴算子A的有限群G的固定不可约表示对应的Dirichlet本征值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，这也应该提供渐近展开中余数的估计。薛定谔算子的谱长期以来一直被解释为量子粒子在与该算子的系数相关的电场和磁场中的能级。这项研究的结果可以用量子粒子在不同能级上的局域化和稳定性来解释。我们计划研究不同态的对称性对相应能级渐近分布的影响。Korteweg-de Vries(KdV)方程和修正的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，我们期望用谱方法得到新的解。

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.