Title:

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Abstract:

(Joint work with Injo Hur)

 The KdV equation has many connections in many different parts of mathematics and physics. For example, it is of critical importance in the inverse spectral theory of the Schrödinger operator, since it describes a way to evolve a Schrödinger operator that keeps its spectrum invariant. This evolution is known as the KdV hierarchy.

Another useful perspective on the inverse spectral theory of the Schrödinger operator is that of the Herglotz m-function. To each Schrödinger operator we associate a holomorphic function from the upper half-plane to itself, such that the limiting behavior of this function on the real line determines the spectrum of the Schrödinger operator. 

We combine these two perspectives on inverse spectral theory, and introduce a version of the KdV hierarchy that applies to all holomorphic functions from the upper half-plane to itself, not just the ones that are associated to a Schrödinger operator. This approach suggests a way to unify a large class of isospectral evolutions for many different operators.

Numerous manifestations of wave localization permeate acoustics, quantum physics, mechanical and energy engineering. It was used in construction of noise abatement walls, LEDs, optical devices, to mention just a few applications. Yet, no systematic methods could predict the exact spatial location and frequencies of the localized waves.

In this talk I will present recent results revealing a new criterion of localization, tuned to the aforementioned questions, and will illustrate our findings in the context of the boundary problems for the Laplacian and bilaplacian, $div A\nabla$,  and (continuous) Anderson and Anderson-Bernoulli models on a bounded domain. Via a new notion of ``landscape" we connect localization to a certain multi-phase free boundary problem and identify location, shapes, and energies of localized eigenmodes. The landscape further provides estimates on the rate of decay of eigenfunctions and delivers accurate bounds for the corresponding eigenvalues, in the range where both classical Agmon estimates and Weyl law may fail.   

Consider the almost Mathieu operator in the case of the critical coupling.

For rational frequencies p/q we obtain power-law bounds of the form 1/q^C

for the widths of the gaps close to the center of the spectrum. It follows that

these gaps remain open for the spectrum in the case of a certain class of Diophantine irrational frequencies.

Abstract: We will consider recent numerous achievements in the area of understanding
the harmonic measure of general domains in\R^n.
These results become possible because of the recent breakthroughs in non-homogeneous
harmonic analysis.