Mathematical Sciences: Toeplitz Operators, Quantum Mechanicsand Mean Oscillation

科伯恩教授将继续他的符号研究 与Berezin-Toeplitz量子化相关的微积分。一 具体设置是具有高斯测度的复n-空间， 在温和的生长和振荡条件下， 符号的奇异性。科伯恩将试图描述 有界汉克尔算子在此设置的平均 符号的振荡。相应的有界性问题 球和多圆盘的哈代空间也将受到攻击。 Coburn希望在这些工作中开发的技术 问题是有用的研究推导的各种 自然发生的代数。 数学研究这一领域的动力 来自于量子力学在经典的，前量子的 纵观世界，物理系统的可观测性是 数字，或者，如果想要跟踪 这些数字对其他量和函数的影响。这是常见 但在微观层面上，它并不起作用。一 需要将函数替换为运算符、线性变换 一个通常是无限维的空间这个替换 数学物理学家称之为量子化。总 这里的思想是，运算符的行为几乎与 他们所做的，只是为了纠正他们所犯的错误。 包含重要信息。在这条奋进的道路上，身体 洞察力经常影响数学，反之亦然。

Professor Coburn will continue his study of the symbol calculus associated with the Berezin - Toeplitz quantization. One specific setting is complex n-space with Gaussian measure, and with moderate growth and oscillation conditions on the singularities of the symbols. Coburn will seek to characterize bounded Hankel operators in this setting in terms of mean oscillation of symbols. The corresponding boundedness problem for the Hardy spaces of the ball and polydisc will also be attacked. Coburn expects the techniques developed in working on these problems to be useful in the study of derivations on various naturally occurring algebras. The impetus for this general area of mathematical research comes from quantum mechanics. In the classical, pre-quantum overview of the world, the observables of a physical system are numbers, or, if one wants to keep track of the dependency of these numbers on some other quantities, functions. This is common sense, but at the microscopic level it doesn't quite work. One needs to replace functions by operators, linear transformations of a space that is usually infinite-dimensional. This replacement is what mathematical physicists call quantization. The general idea is that the operators behave almost like the functions from which they arise, except for correction terms which turn out to contain important information. In this line of endeavor, physical insight informs mathematics as often as the reverse.