Mathematical Sciences: Asymptotic Spectral Problems for     Anharmonic Oscillators
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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.
{'FirstName': 'David', 'LastName': 'Gurarie', 'PI_MID_INIT': None, 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'David Gurarie', 'EmailAddress': 'david.gurarie@case.edu', 'NSF_ID': '000460796', 'StartDate': '06/01/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '126200', 'Text': 'MODERN ANALYSIS'}
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