Abstract: In this course we discuss an approach via extension theory of symmetric operators to define and study selfadjoint Schrödinger operators with delta-type potentials supported on curves and hypersurfaces. Our main ingredient is the abstract method of quasi boundary triples and their Weyl functions - with the help of suitable boundary mappings and analytic properties of Dirichlet-to-Neumann maps we derive a Birman-Schwinger principle and a variant of Krein's resolvent formula, which in turn allow a detailed description of the spectrum and lead to Schatten-von Neumann type estimates for the resolvent differences of Schrödinger operators with delta potentials and the unperturbed free Laplacian.
Abstract: After an introduction into the concept of spectral shift functions, we turn to a variety of applications. Specifically, we will discuss continuity of spectral shift functions with respect to varying the underlying pair of operators and describe how this can be used to characterize the Witten index of a class of non-Fredholm operators. In addition, we will describe applications to a pair of self-adjoint extensions of a given symmetric operator and represent the corresponding spectral shift function in terms of the underlying abstract operator-valued Weyl--Titchmarsh function (an energy parameter dependent Dirichlet-to-Neumann map). This abstract approach will then be illustrated with concrete applications to 2nd order elliptic partial differential operators on n-dimensional domains with compact boundaries and Robin-type boundary conditions and certain classes of multi-dimensional Schrödinger operators.
Abstract: In the lectures we will describe a method to prove Anderson localization that has allowed to solve the arithmetic
spectral transition (from absolutely continuous to singular continuous to pure point spectrum)
problem, in coupling, frequency and phase. We will also present sharp results on the singular continuous regime.
A very captivating question in solid state physics is to determine/understand the hierarchical structure of spectral features of operators describing 2D Bloch electrons in perpendicular magnetic fields, as related to the continued fraction expansion of the magnetic flux. In particular, the hierarchical behavior of the eigenfunctions of the almost Mathieu operators, despite significant numerical studies and even a discovery of Bethe Ansatz solutions has remained an important open challenge even at the physics level.
We will sketch how presented method leads to a complete solution of this problem in the exponential sense throughout the entire localization regime. Namely, we will describe, with very high precision, the continued fraction driven hierarchy of local maxima, and a universal (also continued fraction expansion dependent) function that determines local behavior of all eigenfunctions around each maximum, thus giving a complete and precise description of the hierarchical structure. In the regime of Diophantine frequencies and phase resonances there is another universal function that governs the behavior around the local maxima, and a reflective-hierarchical structure of those, a phenomena not even described in the physics literature.
Abstract: Topological insulators are solid state systems of independent electrons for which the Fermi level lies in a mobility gap, but the Fermi projection is nevertheless topologically non-trivial, namely it cannot be deformed into that of a normal insulator. This non-trivial topology is encoded in adequately defined invariants and implies the existence of surface states that are not susceptible to Anderson localization. After an overview of some experimental results, it will be shown how tools from functional analysis, index theory and non-commutative geometry can be applied to understanding the underlying mathematical structures.
Abstract: The fact, that harmonic waves with different frequencies travel with different speeds is known as dispersion.
Superposition of such waves leads to destructive interference and hence localized wave packages will spread in time yielding decay of the wave package.
Mathematically one is interested in quantifying this dispersive decay and proving that the $L^p$ norm of a solution of a linear evolution equation decays
like a given power of time provided the initial condition is in some other $L^q$. Another way of looking at this fact is to look for some averaged version by
proving that the $L^p$ norm of a solution is in some Lr as a function of time again provided the initial condition is in some $L^q$. These latter types of
estimates are known as Strichartz estimates.
In addition to providing insight to the behavior of solutions of the linear equations they also constitute an important ingredient for investigating the Cauchy problem for nonlinear perturbations of these equations or for proving stability of soliton solutions of associated nonlinear equations. In this short course we will have a glance at this fascinating area of analysis by looking at the Schroödinger equation from quantum mechanics and the associated nonlinear Schrödinger equation.