Mathematical Sciences:  Spectral Asymptotics of Toeplitz andPseudodifferential Operators
{'Code': '03040101', '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}
A continuation of earlier work on the spectral asymptotics                 of pseudodifferential operators and related questions on                        Toeplitz operators will be carried out.                                              Recent expansions associated with pseudodifferential                       operators obtained in a formal sense have been shown, in                        certain cases, to be correct asymptotically.  In effect, one is                 trying to gain information about the trace of functions of an                   operator in terms of some universal approximation which takes                   the form of an integral of the function with respect to a                       measure.  This was the basis for the pioneering work of Weyl                    and Szego in the earlier part of the century on classical                       operators.                                                                           Present theory has produced a Szego expansion for                          operators with a smooth symbol.  The object of the current                      study is to develop expansions for operators with nonsmooth                     symbols.  Such operators are not mathematical artifacts; they                   arise in several areas of applied mathematics.  The approach to                 this question will be based on a method of stationary phase for                 integrals with nonsmooth amplitude function.                                         Work will also be done in seeking a generalization to the                  n-sphere of Szego's refined limit theorem on Toeplitz                           determinants.  Toeplitz matrices enter into these studies                       naturally as one-dimensional discrete analogues of                              pseudodifferential operators, where the multiplier is                           discontinuous.  A crucial inequality in the one-dimensional                     theory would have important implications to harmonic analysis                   in higher dimensions if the correct analogue can be                             established.  Work will be done towards this end.                                    Additional work will be done in an effort to establish a                   functional calculus to support the study of nonselfadjoint                      Toeplitz operators, with emphasis placed on learning how the                    spectra distribute when the multipliers change.
{'FirstName': 'Harold', 'LastName': 'Widom', 'PI_MID_INIT': None, 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Harold Widom', 'EmailAddress': 'widom@math.ucsc.edu', 'NSF_ID': '000303379', 'StartDate': '07/01/1989', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'University of California-Santa Cruz', 'CityName': 'SANTA CRUZ', 'ZipCode': '950641077', 'PhoneNumber': '8314595278', 'StreetAddress': '1156 HIGH ST', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'California', 'StateCode': 'CA', 'CONGRESSDISTRICT': '19', 'CONGRESS_DISTRICT_ORG': 'CA19', 'ORG_UEI_NUM': 'VXUFPE4MCZH5', 'ORG_LGL_BUS_NAME': 'UNIVERSITY OF CALIFORNIA SANTA CRUZ', 'ORG_PRNT_UEI_NUM': None}
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{'Code': '126100', 'Text': 'CLASSICAL ANALYSIS'}
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