Mathematical Sciences: Partial Differential Equations of    Mathematical Physics
{'Code': '03040102', 'Directorate': {'Abbreviation': 'MPS', 'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'Abbreviation': 'DMS', 'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': 'John V. Ryff', 'PO_EMAI': None, 'PO_PHON': None}
Work will be done on problems of partial                                differential equations arising from models of diverse                           physical phenomena.  The work focuses on linear                                 differential operators of three primary types.                                          The first goal will to expand on earler successes                       in studying isospectral Schrodinger operators with                              periodic potentials (if you know that two operators have                        the same spectrum, how are they related?).  Techniques                          should be applicable to general self-adjoint elliptic                           operators with periodic coefficients.  Given that the                           spectrum is known for an operator acting on smooth                              functions on a torus which is a fundamental domain for                          the period lattice, the investigators want to determine                         characteristics of the operator's coefficients.  Two                            important cases worthy of study will be that of the                             Laplace-Beltrami operator and the hamiltonian for an                            electron moving in a magnetic field through a lattice of                        ions which produce the potential.                                                       A second thrust will concentrate on initial-value                       problems for hyperbolic equations defined in domains                            with corners.  The principal investigators succeeded in                         providing conditions along the "edge" for smooth                                solutions of elliptic equations to exist.  In the                               hyperbolic setting singularities not only occur where                           hypersurfaces meet but they also propagate in some                              complex manner.  In particular, gliding rays are                                reflected and transmitted at the edge giving rise to                            cones of rays.  One of the long-term goals of this work                         will be to provide justification to some conjectures                            from the theory of diffraction.                                                         A final area involved concerns approximation in                         solid state physics.  The source of this work lies in                           the study of the Laplace-Beltrami operator.  In a                               setting of exceeding small length scale one arrives at                          wave packets satisfying the time-dependent Schrodinger                          equation up to a precise order.  These packets follow                           trajectories of a hamiltonian.  At least that is                                customarily assumed.  Part of this effort will be                               directed to a careful analysis of the entire model.
[{'FirstName': 'Gregory', 'LastName': 'Eskin', 'PI_MID_INIT': None, 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Gregory Eskin', 'EmailAddress': 'eskin@math.ucla.edu', 'NSF_ID': '000174464', 'StartDate': '07/01/1987', 'EndDate': None, 'RoleCode': 'Co-Principal Investigator'}, {'FirstName': 'James', 'LastName': 'Ralston', 'PI_MID_INIT': 'V', 'PI_SUFX_NAME': 'Jr', 'PI_FULL_NAME': 'James V Ralston', 'EmailAddress': 'ralston@math.ucla.edu', 'NSF_ID': '000089764', 'StartDate': '07/01/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}]
{'Name': 'University of California-Los Angeles', 'CityName': 'LOS ANGELES', 'ZipCode': '900244200', 'PhoneNumber': '3107940102', 'StreetAddress': '10889 WILSHIRE BLVD STE 700', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'California', 'StateCode': 'CA', 'CONGRESSDISTRICT': '36', 'CONGRESS_DISTRICT_ORG': 'CA36', 'ORG_UEI_NUM': 'RN64EPNH8JC6', 'ORG_LGL_BUS_NAME': 'UNIVERSITY OF CALIFORNIA, LOS ANGELES', 'ORG_PRNT_UEI_NUM': None}
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{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
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