Spectral Properties of Elliptic Operators
The spectrum of a natural differential operator, like the Laplacian, on a manifold or in a domain can say a lot about the underlying geometry. For example, the volume can be read off from the spectral asymptotics. One of the traditional methods in spectral theory is to study different functions that are built up from the spectrum of an operator: the zeta-function, the heat trace, the wave trace, etc. For a number of years, in collaboration with Dan Burghelea, Thomas Kappeler, and Patrick McDonald, Lennie Friedlander was involved in the study of the analytic torsion, which is a combination of determinants of the Laplacian acting on forms of different rank.
Eigenvalues of self adjoint operators are central to many applications such as buckling loads, energy levels, and critical values in stability/instability studies. Inner bound approximations can often be obtained by the Rayleigh-Ritz method, or variants including finite elements or Hartree-Fock. The problem of rigorous error estimates is equivalent to finding tight outer bounds. An outer bound method which has considerably greater computational flexibility than others was proposed in the physics literature in the 1960's. General convergence results for this method, now known as the eigenvector-free method or EVF, have been obtained by Marty Greenlee in collaboration with C. A. Beattie of Virginia Tech.
Another interesting direction in spectral theory is the study of constraints on the eigenvalues of certain operators. Lotfi Hermi has established some universal inequalities for the eigenvalues of the Dirichlet Laplacian in a domain.
Operators with periodic coefficients arise naturally in quantum mechanics. Usually, these operators have continuous spectrum that consists of bands. It was observed that the density of states in some high contrast media is concentrated at the right ends of the bands. Lennie Friedlander was able to prove estimates that quantify this statement, and, in his dissertation, former graduate student Jeff Selden established a precise asymptotics of the integrated density of states function near the right ends of the bands.
Group Members
Former Graduate Student
- Jeff Selden, now at the University of Northern Iowa.