Mathematical Sciences: The Schrodinger Equation

这个项目的基本目标是阐明量子力学系统的数学和物理。三个基本领域被挑选出来进行研究：量子理论中的磁场，约束系统的量子理论，以及嵌入的本征值。感兴趣的磁场问题涉及大磁场和哈密顿算子(特别是其谱)在此极限下的极限行为的存在性。自旋为零的粒子和自旋为1/2的粒子将被考虑。在经典力学中，位形空间中的约束可以通过一个极限过程来施加，当取此极限时，位形空间中系统的轨道接近于受约束系统的轨道。在量子力学中，这一点远非明朗。这项研究项目旨在澄清这一情况。许多研究人员已经证明，量子力学哈密顿量的嵌入本征值是不稳定的。它们往往在扰动下消失。这个项目的一部分是理解防止这些本征值消失的各种扰动。当一个没有自旋的量子力学粒子与磁场相互作用时，它仅仅由于处于非零场中而获得零点能量。这种零点能量对于大场来说是很大的，并且具有排除具有合理能量的粒子的效果。因此，当磁场较大时，磁场为零的区域的几何形状非常重要。这种情况将在自旋为零的情况下以及自旋为1/2的情况(电子)中进行研究，后者要困难得多。在量子力学中，将粒子限制在表面上可能是一个困难的过程，因为测不准原理将预测垂直于表面的无限动量波动。然而，受约束的系统似乎存在。这个问题也将得到调查。最后，如果一个粒子通向自由的通道是开放的(即使它需要通过障碍物进行隧道传输)，那么只有在非常特殊的情况下，粒子才会被束缚。这个项目的一部分目的是了解那些仍将束缚粒子的障碍物的结构。

Abstract Herbst The basic objective of this project is to shed light on the mathematics and physics of quantum mechanical systems. Three basic areas are singled out for research: magnetic fields in quantum theory, the quantum theory of constrained systems, and embedded eigenvalues. The magnetic field problems of interest involve large magnetic field and the existence of a limiting behavior of the Hamiltonian operator (in particular its spectrum) in this limit. Spin zero as well as spin 1/2 particles will be considered. In classical mechanics, constraints in configuration space can be imposed by a limiting procedure, and as this limit is taken the orbit of the system in configuration space approaches the orbit of the constrained system. In quantum mechanics, this is far from clear. This research project is aimed at clarifying the situation. Many researchers have shown that embedded eigenvalues of quantum mechanical Hamiltonians are unstable. They tend to disappear under perturbation. Part of this project is to understand the manifold of perturbations which prevent the disappearance of these eigenvalues. When a spinless quantum mechanical particle interacts with a magnetic field, it acquires a zero point energy just by virtue of its being in a non-zero field. This zero point energy is large for large field and has the effect of excluding particles with reasonable energies. Thus, the geometry of the regions where the magnetic field is zero is very important when the field is large. This situation will be investigated in the spin zero case as well as the spin 1/2 case (electrons) which is much more difficult. Confining a particle to a surface in quantum mechanics can be a difficult procedure since the uncertainty principle would predict infinite momentum fluctuations perpendicular to the surface. Yet constrained systems seem to exist. This problem will also be investigated. Finally, if a channel to freedom is open for a particle (even if it requires tunneling through a ba rrier), then only in very special situations will a particle be bound. It is the purpose of part of this project to understand the structure of those barriers which will nevertheless bind a particle.