| Date: | March 8, 2019 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Dr. Selim Sukhtaiev, Rice University |
| Title: | Localization for Anderson Models on Metric and Discrete Tree Graphs |
| Abstract: | In this talk I will discuss spectral and dynamical localization for several Anderson models on metric and discrete radial trees. The localization results are obtained on compact intervals contained in the complement of discrete sets of exceptional energies. All results are proved under the minimal hypothesis on the type of disorder: the random variables generating the trees assume at least two distinct values. This level of generality, in particular, allows us to treat radial trees with disordered geometry as well as Schr\"odinger operators with Bernoulli-type singular potentials. This is based on joint work with D. Damanik and J. Fillman.
JOINT WITH THE GROUPS AND DYNAMICS SEMINAR |
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| Date: | March 22, 2019 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | P. Kuchment, Texas A&M |
| Title: | Non-degeneracy of spectral edges for periodic operators (Joint work with Ngoc Do (U. Arizona) and F. Sottile (TAMU)) |
| Abstract: | An old problem in mathematical physics deals with the structure of the dispersion relation of the Schroedinger operator -\Delta+V(x) in R^n with periodic potential near the edges of the spectrum, i.e. near extrema of the dispersion relation. A well known and widely believed conjecture says that generically (with respect to perturbations of the periodic potential) the extrema are attained by a single branch of the dispersion relation, are isolated, and have non-degenerate Hessian. In particular, the notion of effective masses hinges upon this conjecture.
The progress in proving this conjecture has been rather slow. It is natural to try to look at discrete problems, where the dispersion relation is (in appropriate coordinates) an algebraic, rather than analytic, variety. Moreover, such models are often used for computation in solid state physics (the so called tight binding model). However, counterexamples exist that show that the genericity can fail in discrete situation.
The authors consider the case of a general periodic discrete operator depending polynomially on some parameters and prove that the non-degeneracy of extrema either fails for all values of parameters, or holds for all values except of a proper algebraic subset. Thus, finding a single point in the parameter space where the non-degeneracy holds implies that it holds generically.
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| Date: | April 5, 2019 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Burak Hatinoglu, Texas A&M University |
| Title: | Mixed Data in Inverse Spectral Problems for the Schroedinger Operators |
| Abstract: | In this talk, we consider the Schroedinger operator, Lu = -u''+qu on (0,pi)
with a potential q\in L^1(0,\pi). Borg's theorem says that q can be uniquely recovered
from two spectra. By Marchenko, q can be uniquely recovered from spectral measure. After
recalling some results from inverse spectral theory of one dimensional Schroedinger operators,
we will discuss the following problem: Can q be recovered from support of spectral measure,
which is a spectrum, and partial data on another spectrum and the set of pointmasses of the
spectral measure? |
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| Date: | April 12, 2019 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Dr. Lennie Friedlander, University of Arizona |
| Title: | The Dirichlet-to-Neumann operator for quantum graphs |
| Abstract: | TBA |
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