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.
