It was shown by C. L. Siegel (1929) that the eigenvalues of the vibrating membrane problem has no non-trivial multiplicities. In this talk we consider the eigenvalues of the vibrating clamped plate problem. This is a fourth order problem. We show that its eigenvalues have multiplicity at most six. The proof is based on a new recursion formula for a Bessel-like function and on Siegel-Shidlovskii Theory. If time permits we also consider the problem of determining the density of the nodal sets of a clamped plate.
Abstract: Perturbation theoryworks well for the the discrete spectrum below the essential spectrum. Whathappens if a parameter of a quantum system is tuned in such a way that abound state energy (e.g. the ground state energy) hits the bottom of theessential spectrum? Does the eigenvalue survive, i.e., the correspondingeigenfunction stays $L^2$, or does it dissolve into the continuumenergies?
ABSTRACT:
We study sufficient conditions for the absence of positive eigenvalues of magnetic Schroedinger operators in R^n. In our main result we prove the absence of eigenvalues above certain threshold energy which depends explicitly on the magnetic and electric field. A comparison with the examples of Miller-Simon shows that our result is sharp as far as the decay of the magnetic field is concerned.
The talk is based on a joint work with Silvana Avramska-Lukarska and Dirk Hundertmark.
There is a rich history of studying dynamical systems through the lens of operator algebras, and particularly through C*-algebras. For instance, in the work of Giordano, Matui, Putnam and Skau, C*-algebras were used as a key tool for classifying Cantor minimal $\mathbb{Z}^d$ systems up to various notions of orbit equivalence. Another successful study was conducted by Cuntz and Krieger, where subshifts of finite type (SFTs) are interpreted through C*-algebras of directed graphs, and invariants studied in symbolic dynamics naturally arise from these C*-algebras. Read more about Analysis Seminar: Adam Dor-On (Copenhagen) "Operator algebras for subshifts and random walks"
Charged domain walls are a type of transition layers in thin ferromagnetic films which appear due to global topological constraints. The underlying micromagnetic energy is determined by a competition between a diffuse interface energy and the long-range magnetostatic interaction. The underlying model is non-convex and vectorial. In the macroscopic limit we show that the energy Γ-converges to a limit model where jump discontinuities of the magnetization are penalized anisotropically. In particular, we identify a supercritical regime which allows for tangential variation of the domain walls.
Consider maps $u:R^n\to R^k$ with values constrained in a fixed submanifold, and minimizing (locally) the energy $E(u)=\int W(
abla u)$. Here $W$ is a positive definite quadratic form on matrices. Compared to the isotropic case $W(
abla u)=|
abla u|^2$ this may look like a harmless generalization, but the regularity theory for general $W$'s is widely open. I will explain why, and describe results with Andres Contreras on a relaxed problem, where the manifold-valued constraint is replaced by an integral penalization.
Zoom link:
We show short-time existence and uniqueness for the surface diffusion flow with a nonlocal forcing of elastic type. We also establish long-time existence and asymptotic behavior for a suitable class of strictly stable initial data. To the best of our knowledge these are the first rigorous results for a surface diffusion evolution equation with elastic stress and without curvature regularization.
In 1966 V. Arnold made an astonishing discovery: the incompressible Euler equations describe Riemannian geodesics on the infinite-dimensional “Lie group" of volume-preserving diffeomorphisms. This discovery led to geometric hydrodynamics – a field that today encompasses many equations of mathematical physics, information theory, shape analysis, etc. In this talk I shall address the infinite-dimensional manifold and group structures assigned to spaces of diffeomorphisms. Having such structures in place often enable out-of-the-box local existence and uniqueness results.
