Mathematical Sciences: Schrodinger Operators and Elliptic Eigenvalue Problems with an Indefinite Weight Function

本文研究了薛定谔算子的谱理论和带不定权函数的椭圆边值问题。这种特征值问题在非线性偏微分方程的研究中自然出现，在数学物理和应用数学中具有广泛的意义。在物理学和工程学中，它们出现在传输理论、激光、晶体着色、流体力学和反应扩散中。拉皮德斯教授将研究半线性椭圆方程以及与之相关的不确定权值的特征值问题。椭圆型问题设置在有界区域上，具有Dirichlet或Neumann边界条件。他还将考虑具有离散谱的无界域上的薛定谔算子。他将研究特征值和特征函数，特别强调经典渐近公式的扩展和剩余估计，特征值的分布，反问题，以及特征函数的衰减和振荡。

This research treats spectral theory of Schrodinger operators and elliptic boundary value problems with indefinite weight functions. Such eigenvalue problems occur naturally in the study of nonlinear partial differential equations and are of broad interest in mathematical physics and applied mathematics. In physics and engineering they arise, for instance, in transport theory, lasers, crystal coloration, hydrodynamics, and reaction-diffusion. Professor Lapidus will conduct research on semilinear elliptic equations and the related topic of eigenvalue problems with indefinite weight. The elliptic problems are set on bounded domains with Dirichlet or Neumann boundary conditions. He will also consider Schrodinger operators on unbounded domains with discrete spectrum. He will investigate both eigenvalues and eigenfunctions, with special emphasis on extensions of classical asymptotic formulas and remainder estimates, distribution of eigenvalues, inverse problems, and decay and oscillation of the eigenfunctions.