Mathematical Sciences: Spectral Problems and Inverse        Spectral Problems
{'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}
This project focuses on research concerned with three                      related areas of mathematical analysis.  In each case the work                  continues earlier investigations of a similar nature.                                The first objectives derive from efforts to apply inverse                  scattering results for selfadjoint ordinary differential                        operators.  These operators arise naturally in the analysis of                  nonlinear wave equations which are known to be integrable.                      Although explicit solutions can be obtained by inverse methods,                 the long time behavior of solutions is not well understood, and                 is far from complete.  The difficulty lies in the sign of the                   group velocity which does not stay fixed independent of energy,                 so disturbances move out in both directions.  This means that                   free solutions are not good approximations for the full                         scattering solutions of the associated linear problem.  A good                  parametrix for the problem with general initial data will be                    sought.                                                                              Work will also be carried out on matrix inversion problems.                The connection with differential equations is made through recent               discoveries that diagonalization of self-adjoint matrices can be                viewed in the context of integrable Hamiltonian systems.  It was                further observed that the QR, LU and Cholesky algorithms for                    nonsymmetric matrices also fit into this context.  Moreover, the                computational potential for this approach is most promising.                    Work will continue in studying the long term asymptotics of the                 Rayleigh shifted QR algorithm for nonsymmetric matrices.                             A third topic of concern deals with spectral problems for                  Schrodinger operators.  An important question arising in the                    theory of crystal coloration revolves around the existence of                   eigenvalues of a perturbed Schrodinger operator.  More precisely,               one is interested in determining the extent to which eigenvalues                of a perturbed operator can materialize within gaps in the                      spectrum of the original operator.  The phenomenon is known to                  occur, the issue is to find suitable information on the                         distribution of the spectrum
{'FirstName': 'Percy', 'LastName': 'Deift', 'PI_MID_INIT': 'A', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Percy A Deift', 'EmailAddress': 'deift@cims.nyu.edu', 'PI_PHON': '2129983288', 'NSF_ID': '000193635', 'StartDate': '07/15/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
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