无限域上薛定谔算子的解、谱和动力学
Solutions, Spectrum, and Dynamica for Schrödinger Operators on Infinite Domains
摘要：
设H是定义在R^d中无界区域D上的具有Dirichlet边界条件的薛定谔算子（D可以特别地等于R^d）。设u（x，E）是薛定谔方程（H-E）u（x，E）=0的解，B_R表示一个半径为R的球.我们证明了当R趋于无穷远时，这类解的L^2范数u（x，E）{L^2（B_R \cap D）}的增长率与算子H的谱测度的连续性之间的关系.这些结果自然会导致新的标准，用于识别各种光谱特性。我们还证明了广义本征函数的L^2范数的增长率、谱测度的维数性质和相应量子系统的动力学性质之间的新的基本关系。我们应用这些结果来研究一些特殊的薛定谔算子的输运性质。
英文摘要：
Let H be a Schrödinger operator defined on an unbounded domain D in R^d with Dirichlet boundary conditions (D may equal R^d in particular). Let u(x,E) be a solution of the Schrödinger equation (H-E)u(x,E)=0, and let B_R denote a ball of radius R centered at zero. We show relations between the rate of growth of the L^2 norm \|u(x,E)\|_{L^2(B_R \cap D)} of such solutions as R goes to infinity, and continuity properties of spectral measures of the operator H. These results naturally lead to new criteria for identification of various spectral properties. We also prove new fundamental relations berween the rate of growth of L^2 norms of generalized eigenfunctions, dimensional properties of the spectral measures, and dynamical properties of the corresponding quantum systems. We apply these results to study transport properties of some particular Schrödinger operators.
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