Mathematical Sciences: Asymptotic Spectral Problems for Anharmonic Oscillators

数学物理的微分方程提供了 自然框架和语言来模拟各种物理 经典力学、连续介质力学、量子力学 在量子力学中，基本对象是 薛定谔算符，或量子哈密顿量。 各营办商 在数学和物理的许多其他领域出现，特别是在 波现象的研究：电磁学、声学、气体或 流体动力学或弹性。 考虑到这样一个算子， 特征值和特征向量 很危险 在量子力学中，光谱对应于 原子发射或吸收辐射的能级，或 分子。 在波动现象中，本征数据对应于 系统的基本振动模式。 Gurarie教授的研究涉及到 薛定谔算子和更一般的椭圆算子。 他 这一主题的早期工作涉及闭合问题， 自付财产，建筑估计， 预解式和热半群核，估计和渐近性 特征值和特征函数。 他建议继续调查 更微妙的方面的光谱理论;即 正/逆问题，涉及“几何”的 一个微分算子的系数的频谱。 在 实际上，逆问题被解释为 根据操作者的测量结果确定系统的物理性质， 江西篇章 Gurarie教授研究了 量子力学谐振子，专注于 谱的渐近性 他建议推广这些结果 更广泛的类别的运营商和扰动，使用新的 想法和技术。 这项研究，其中包括一个长期的 程序的光谱和几何，将产生一个改进的 逆问题的知识。

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.