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Research Spending & Results

Award Detail

Doing Business As Name:University of Delaware
  • Constanze D Liaw
  • (302) 831-2964
Award Date:10/19/2017
Estimated Total Award Amount: $ 129,000
Funds Obligated to Date: $ 85,025
  • FY 2017=$85,025
Start Date:07/01/2017
End Date:06/30/2020
Transaction Type:Grant
Awarding Agency Code:4900
Funding Agency Code:4900
CFDA Number:47.049
Primary Program Source:040100 NSF RESEARCH & RELATED ACTIVIT
Award Title or Description:Finite Rank Perturbations and Model Theory
Federal Award ID Number:1802682
DUNS ID:059007500
Parent DUNS ID:059007500
Program Officer:
  • Tomek Bartoszynski
  • (703) 292-4885

Awardee Location

Street:210 Hullihen Hall
Awardee Cong. District:00

Primary Place of Performance

Organization Name:University of Delaware
Cong. District:00

Abstract at Time of Award

Many physical systems are modeled by differential operators. One way of describing a system's long-term behavior is through spectral theory, which includes the finding of frequencies naturally exhibited by the system. Imagine a vibrating string or beam of fixed length. Clearly, its frequency (think "sound") depends on how its ends (aka boundaries) are clamped down or otherwise restricted. In general, the spectrum of a differential operator and with it the properties of a physical system can change drastically when the conditions imposed on the boundary are changed. In many cases, we know the complete spectrum for one set of boundary conditions. From there, we can gain knowledge about the system under other conditions via the theory of so-called finite rank perturbations. The primary goal of the project is to systematically study spectral theory of finite rank unitary perturbations by tightening the relationship to corresponding functional models. The rank one setting is reasonably well-understood, while a general treatment of the finite rank problem presented several digressions. Initially, non-cyclic unitary unperturbed operators posed an issue. This problem is now resolved. The general problem involves matrix-valued Herglotz and Cauchy-type transforms. The study of these transforms as part of this project will provide insight into finite rank perturbation theory. A regularization of the so-called exterior Cauchy transform (as was done in the rank one setting) is planned. Further investigations will be made into the difficult case of infinite rank perturbations with defect operators in the trace class. The analogous self-adjoint setting, as well as concrete applications to differential operators, will also be investigated.

Publications Produced as a Result of this Research

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Kostadinova, E.G. and Guyton, F. and Cameron, A. and Busse, K. and Liaw, C. and Matthews, L.S. and Hyde, T.W. "Transport properties of disordered two-dimensional complex plasma crystal" Contributions to Plasma Physics, v.58, 2018, p.. doi:10.1002/ctpp.201700111 Citation details  

Kostadinova, E. G. and Busse, K. and Ellis, N. and Padgett, J. and Liaw, C. D. and Matthews, L. S. and Hyde, T. W. "Delocalization in infinite disordered two-dimensional lattices of different geometry" Physical Review B, v.96, 2017, p.. doi:10.1103/PhysRevB.96.235408 Citation details  

Kelly, Jessica Stewart and Liaw, Constanze and Osborn, John "Moment representations of exceptional X 1 orthogonal polynomials" Journal of Mathematical Analysis and Applications, v.455, 2017, p.. doi:10.1016/j.jmaa.2017.05.037 Citation details  

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