We consider discrete quasi-periodic long range operators with Liouvillean frequency. First, based on generalized Gordon type argument, we show that they can be approximated by a sequence of finite range operators which have no point spectrum for any phase. On the other hand, we show that when the potential for the dual model is small, then they can be approximated by a sequence of long range operators which have at least one eigenvalue for each phase in a set of full measure.
We prove that the resolvent set of any (possibly singular)
almost periodic Jacobi operator is characterized as the set of all
energies whose associated Jacobi cocycles induce a dominated splitting.
This extends a well-known result by Johnson for Schrödinger operators.
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.
The classical Huaxin Lin's theorem shows that the distance from a matrix A to the set of normal matrices can be estimated in terms of its self-commutator [A,A*]. We obtain a quantitative version of this theorem, "optimal" with respect to the power of self-commutator. Under certain assumptions on A, our approach can be extended to the case of general bounded operators in Hilbert spaces and to elements of C*-algebras of real rank zero. The results are joint with Professor Yuri Safarov from King's College London.
In 1983 Dan Voiculescu used a family of unitary matrices, now
known as "Voiculescu's Unitaries," to provide the first counter-example to
an old conjecture of Halmos regarding "almost commuting" matrices. Later,
Ruy Exel and Terrry Loring used "Voiculescu's Unitaries" in an elementary
and elegant proof to provide another counter-example on "almost commuting"
matrices. In this talk, we present two new counter-examples using
"Voiculescu's Unitaries." The talk should be accessible to anyone with
knowledge of basic real analysis and linear algebra.