Nodal count of eigenfunctions as an index of instability

Speaker: 

G. Berkolaiko

Institution: 

Texas A&M

Time: 

Thursday, March 13, 2014 - 2:00pm

Location: 

RH 340P

Zeros of vibrational modes have been fascinating physicists for several centuries. Mathematical study of zeros of eigenfunctions goes back at least to Sturm, who showed that, in dimension d=1, the n-th eigenfunction has n-1 zeros. Courant showed that in higher dimensions only half of this is true, namely zero curves of the n-th eigenfunction of the Laplace operator on a compact domain partition the domain into at most n parts (which are called "nodal domains").

 It recently transpired that the difference between this upper bound and the actual value can be interpreted as an index of instability of a certain energy functional with respect to suitably chosen perturbations. We will discuss two examples of this phenomenon: (1) stability of the nodal partitions of a domain in R^d with respect to a perturbation of the partition boundaries and (2) stability of a graph eigenvalue with respect to a perturbation by magnetic field. In both cases, the "nodal defect" of the eigenfunction coincides with the Morse index of the energy functional at the corresponding critical point. We will also discuss some applications of the above results.
 
The talk is based on joint works with R.Band, P.Kuchment, H.Raz, U.Smilansky and T.Weyand.

Phase transition for Quasi-Periodic Schr\"odinger Operators

Speaker: 

Qi Zhou

Institution: 

Paris 6

Time: 

Thursday, March 6, 2014 - 2:00pm

Location: 

RH 340P

In this talk, we will talk about two phase transiton results for quasi-Periodic Schr\"odinger Operators. 

For continuous Sch\"odinger operators with large analytic quasi-periodic 
potentials of two frequencies, we obtain the exact power-law for phase transition in energy.

For the almost Mathieu operator with any fixed frequency, we locate 

the point where phase transition from  singular continuous spectrum to pure point spectrum takes place, 

which solves Jitomirskaya's conjecture

Anderson Localization for the almost Mathieu operator for general irrational frequency

Speaker: 

Wencai Liu

Institution: 

Fudan University, visiting UCI

Time: 

Thursday, February 27, 2014 - 2:00pm

The key to set up Anderson Localization is to estimate the Green function. In this talk, I will introduce two ways to estimate the Green function.

One is the way of Harmonic analysis(based on Bourgain's book," Green's function  estimates for lattice Schrodinger operators and application"). The other way is by Jitomirskaya (based on two papers in the Annals of Math( 1999 and 2009). In the end, I will give an

extended result on Anderson Localization (this is a joint work with Xiaoping Yuan).

The bootstrap multiscale analysis and localization for multi-particle continuous Anderson Hamiltonians

Speaker: 

Son Nguyen

Institution: 

University of Missouri

Time: 

Thursday, February 13, 2014 - 2:00pm

Location: 

RH 340P

We extend the bootstrap multiscale analysis to multi-particle continuous Anderson Hamiltonians, obtaining Anderson localization with finite multiplicity of eigenvalues, a strong form of dynamical localization, and decay of eigenfunction correlations. (Joint work with Abel Klein)

Eigenfunctions on billiard tables, III

Speaker: 

Hamid Hezari

Institution: 

UCI

Time: 

Thursday, March 20, 2014 - 2:00pm to 3:00pm

Location: 

rh340P

 

Eigenfunctions of the Laplacian on a bounded domain represent the modes of vibration of a vibrating drum. The behavior of these eigenfunctions is closely related to the behavior of the underlying dynamical system of the billiard table. In this talk I first give a brief exposition on this relation and then I talk about the boundary traces of eigenfunctions and a recent joint work with Han, Hassell and Zelditch.

 

Big frequency cascades in the nonlinear Schrödinger evolution

Speaker: 

James Colliander

Institution: 

University of Toronto

Time: 

Thursday, January 23, 2014 - 2:00pm

Location: 

RH 340P

 I will outline a construction of an exotic solution of the nonlinear
Schrödinger equation that exhibits a big frequency cascade. Recent advances
related to this construction and some open questions will be surveyed.

Eigenfunctions on billiard tables, II

Speaker: 

Hamid Hezari

Institution: 

UCI

Time: 

Thursday, January 16, 2014 - 2:00pm

Location: 

RH 340P

Eigenfunctions of the Laplacian on a bounded domain represent the modes of vibration of a vibrating drum. The behavior of these eigenfunctions is closely related to the behavior of the underlying dynamical system of the billiard table. In this talk I first give a brief exposition on this relation and then I talk about the boundary traces of eigenfunctions and a recent joint work with Han, Hassell and Zelditch.

Threshold effects of the two and three-particle Schroedinger operators on lattices

Speaker: 

Saidakhmat Lakaev

Institution: 

University of California, Davis and Samarkand State University, Uzbekistan

Time: 

Monday, May 19, 2014 - 2:00pm

Host: 

Svetlana Jitomirskaya

Location: 

TBD

The Hamiltonians of two and three particles moving on d-dimensional lattice and interacting via pairwise short-range potentials are studied.
The following new results are established:
(i).The existence of eigenvalues for the two-particle Shr\"odinger operators depending on the quasi-momentum.
(ii). Infiniteness the number of eigenvalues(Efimov's effect) of the three-particle Shr\"odinger operators
for the zero value of quasi-momentum and its finiteness for the non-zero values of the quasi-momentum.
(iii).The corresponding asymptotics for the number of eigenvalues.

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