Mathematical Sciences: Asymptotic Spectral Problems for Anharmonic Oscillators

数学物理的微分方程为经典力学、连续介质力学、量子理论等各种物理系统的建模提供了自然的框架和语言。在量子力学中，基本对象是薛定谔算符，或量子哈密顿算符。这样的算子出现在数学和物理的许多其他领域，特别是在波动现象的研究中：电磁学，声学，气体或流体动力学，或弹性。给定这样一个算子，研究它的谱——特征值和特征向量——是至关重要的。在量子力学中，光谱对应于原子或分子发射或吸收的辐射的能级。在波动现象中，本征数据对应于系统的基本振动模态。Gurarie教授的研究涉及薛定谔算子的谱和更一般的椭圆算子。他在这方面的早期工作涉及闭包问题、自伴随性质、解半群核和热半群核的估计的构造、特征值和特征函数的估计和渐近性。他建议研究光谱理论更微妙的方面；也就是将微分算子系数的“几何”与谱联系起来的正/反问题。在实际中，反问题被解释为从操作员的频谱中确定系统的物理性质。Gurarie教授研究了量子力学谐振子的扰动，重点是谱的渐近性。他建议使用新颖的思想和技术，将这些结果扩展到更广泛的算子和微扰类别。这项研究包括一个关于光谱和几何的长期计划，将提高人们对反问题的认识。

Differential equations of mathematical physics provide the natural framework and the language to model a variety of physical systems in classical mechanics, continuum mechanics, quantum theory, etc. In quantum mechanics the basic object is the Schrodinger operator, or quantum Hamiltonian. Such operators arise in many other areas of mathematics and physics, notably in the study of wave phenomena: electromagnetism, acoustics, gas or fluid dynamics, or elasticity. Given such an operator, the study of its spectrum -- the eigenvalues and eigenvectors -- is critical. In quantum mechanics, the spectrum corresponds to the energy levels of emitted or absorbed radiation of atoms or molecules. In wave phenomena, the eigendata corresponds to the basic modes of vibration of the system. Professor Gurarie's research deals with the spectrum of Schrodinger operators and more general elliptic operators. His earlier work in this subject concerned problems of closure, the self-adjoiint property, construction of estimates for the resolvent and heat semigroup kernel, estimates and asymptotics of eigenvalues and eigenfunctions. He proposes to investigate still more subtle aspects of the spectral theory; namely the direct/inverse problems that relate the "geometry" of the coefficients of a differential operator to the spectrum. In practical terms, the inverse problem is interpreted as determining physical properties of the system from the operator's spectrum. Professor Gurarie has investigated perturbations of the quantum mechanical harmonic oscillator, focusing on the asymptotics of the spectrum. He proposes to extend these results to wider classes of operators and perturbations, using novel ideas and techniques. This study, which comprises a long range program on spectra and geometry, would yield an improved knowledge of inverse problems.