Grants and Contracts per year
Grants and Contracts Details
Description
The principal investigator will study inverse spectral and scattering theory in three contexts:
(1) the spectral theory of nilpotent Lie groups and compact nilmanifolds, (2) the inverse
resonance problem for exterior domains and scattering manifolds, and (3) inverse scattering
for very singular potentials with applications to nonlinear dispersive equations.
In joint work with Ruth Gornet, the principal investigator will attack a fundamental
conjecture in inverse spectral theory by studying trace formulas on twostep compact nil
manifolds. Twostep nilpotent Lie groups G and their quotients by cocompact discrete
subgroups carry two natural geometric and analytic structures: the Carnot geometry associ
ated with the subLaplacian, and the Riemannian geometry associated with the Laplacians
for leftinvariant metrics. We will study the spectral theory and \geometric optics" associ
ated with the Riemannian Laplacians and subLaplacians in order to derive trace formulas
for nilmanifolds. Using these trace formulas, the principal investigator and Gornet will study
the relationship between length spectrum and Laplace spectrum.
The principal investigator will study the geometry of scattering resonances. Scattering
resonances are discrete data associated to the wave equation on a complete, noncompact
Riemannian manifold or an exterior domain in Rn. Like the eigenvalues of the Laplacian on
a compact manifold, they are thought to carry geometric information. To elucidate the geo
metric content of resonances, the principal investigator will continue his work with Carolyn
Gordon and Dorothee Schueth to construct pairs and families of nonisometric manifolds and
exterior domains with the same scattering resonances, with particular attention to exterior
domains in the plane. In joint work with Paul Yang, the principal investigator will consider
the inverse resonance problem for conformally Euclidean manifolds.
In joint work with Mikhail Shubin, Thomas Kappeler, and Peter Topalov, the princi
pal investigator will also study inverse scattering theory for SchrÄodinger operators on the
line with highly singular potentials and its application to wellposedness questions for the
Kortewegde Vries equation. Building on work in progress on the Miura map on the line,
the principal investigator and these collaborators will also study wellposedness questions
for the modi¯ed Kortewegde Vries equation.
Eigenvalues and scattering resonances are measurable quantities in electromagnetic and
acoustic wave propagation. The work carried out here will elucidate the geometric content of
eigenvalues and resonances and suggest what geometric quantities can and cannot be recon
structed from such measurements. The inverse scattering method applies to a large family
of dispersive equations; the principal investigator's methods used for SchrÄodinger equation
on the line with very singular potentials should extend to other inverse scattering prob
lems relevant to this area. The study of spectral theory and trace formulas on nilmanifolds
will improve our understanding of analysis and geometry on nilpotent Lie groups and their
quotients, and illuminate the connection between spectral and geometric invariants.
Grant activities will impact human resources through graduate training and ongoing col
laborations supported by the grant. In related e®orts, the principal investigator and Carolyn
Gordon will organize yearly conferences on inverse spectral geometry which will support doc
toral students and recent doctoral graduates; these conferences will bring together experts in
scattering theory and di®erential geometry in order to catalyze developments in the inverse
resonance problem.
Status  Finished 

Effective start/end date  6/1/04 → 5/31/08 
Funding
 National Science Foundation: $153,316.00
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Projects
 1 Finished

Inverse Problems in Geometry and Partial Differential Equations
6/1/04 → 5/31/08
Project: Research project