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2025

Existence and analytic behavior of Jost solutions for Schrödinger operators on the Discrete Line with varying spectral multiplicity

at The Americas-Europe Conference in Ciudad de México, México

youtube.com/watch?v=rDTbWOXZePI

Abstract: We study scattering theory for a class of matrix-valued Schrödinger operators on a one-dimensional lattice. In our model, the free Hamiltonian is given by a variation of the discrete Laplacian, and we consider a matrix potential satisfying a condition of sufficient decay. We prove the existence of Jost solutions for our model and examine their analytic properties.

A Levinson-type Theorem for Scattering Matrices with Varying Dimension for Schrödinger Operators on the Discrete Line

at the Math4Q Conference in Mérida, México

youtube.com/watch?v=uv0t5j9zlQg

Abstract: We study scattering theory for a class of matrix-valued Schrödinger operators on a one-dimensional lattice. In our model, the free Hamiltonian is given by a variation of the discrete Laplacian, and we consider a matrix potential satisfying a condition of sufficient decay. We prove a Levinson-type Theorem for our model, relating the scattering data to the number of bound and half-bound states. Since, in our model, the dimension of the scattering matrix depends on the energy, this requires a delicate analysis of the analytic behavior of the scattering matrix in the thresholds.

Outgoing monotone geodesics of standard subspaces

at the Lie-Group seminar of the Friedrich-Alexander-University Erlangen-Nürnberg, Germany

Abstract: Using a real version of the Lax–Phillips Theorem, we provide a normal form for outgoing monotone geodesics in the set Stand(H) of standard subspaces of a complex Hilbert space H. As the modular operators of a standard subspace are closely related to positive Hankel operators, our results are obtained by constructing some explicit symbols for positive Hankel operators. We also describe which of the monotone geodesics in Stand(H) arise from the unitary one-parameter groups described in Borchers’ Theorem and provide explicit examples of monotone geodesics that are not of this type.

Stationary Scattering Theory with Varying Spectral Multiplicity on the Discrete Line

at the mathematical physics seminar of the TU Braunschweig, Germany

2024

Algebraic Quantum Field Theory: Introduction and Applications of the Lax-Phillips Theorem

at the Mathematics Colloquium of Brigham Young University, Provo, Utah, USA

math.byu.edu/events/colloquium-jonas-schober

Abstract: Algebraic quantum field theory is one of the main attempts to provide a formal mathematical framework for quantum field theory. We give an introduction to algebraic quantum field theory, whose core idea is to assign to every space-time region an algebra of local observables. These algebras, who are assumed to be von Neumann algebras, should obey the Haag-Kastler axioms, describing how the algebras assigned to different space-time regions should interact with each other. We show how to translate the Haag-Kastler axioms from von Neumann algebras to the simpler concept of standard subspaces. This leads us to the investigation of one-parameter semigroups of unitary endomorphisms of standard subspaces, which we analyze through the lens of a real version of the classical Lax-Phillips Theorem, originally developed in the context of scattering theory.

The scattering matrix and a Levinson Theorem for a class of matrix-valued discrete Schrödinger operators

at the Lie-Group seminar of the Friedrich-Alexander-University Erlangen-Nürnberg, Germany

Abstract: On the vector-valued l²-space over the integers we consider a free Hamiltonian that is a variation of the discrete Laplacian and study scattering theory for the perturbation of this operator by a matrix-valued potential. We prove the existence of the Jost-solutions and use them to construct the scattering matrix, whose dimension in our model depends on the spectral parameter. The scattering matrix and its analytic properties are then used to formulate a Levinson type Theorem.

One-parameter semigroups of unitary endomorphisms of standard subspaces and reflection positivity

at the conference CUWB-III: Differential and difference equations in models of physics and biology

youtube.com/watch?v=RR1BeUHpUMk

Abstract: Motivated by the Haag-Kastler theory of local observables in Quantum Field Theory, one is interested in unitary endomorphisms of standard subspaces. This talk focuses on one-parameter semigroups of these unitary endomorphisms and shows how they relate to reflection positivity, Hankel operators, and Pick functions.

2023

Standard subspaces, reflection positivity, Pick functions and how to pass from one to the other

at the Lie-Group seminar of the Friedrich-Alexander-University Erlangen-Nürnberg, Germany

Abstract: Motivated by the Haag-Kastler theory of local observables in Quantum Field Theory, one is interested in unitary endomorphisms of standard subspaces. Classical results in this context are given in the works of Borchers and Longo/Witten.
In my PhD project I focus on one-parameter semigroups of these unitary endomorphisms. In the context of Borchers theorem a natural approach is to pass from one-parameter semigroups of unitary endomorphisms of standard subspaces to reflection positive unitary one-parameter groups, while in the context of Longo/Witten one thinks of these one-parameter semigroups as one-parameter semigroups of inner functions which are infinitesimally generated by Pick functions. In this talk I will show how to pass from one picture to the other.

2022

One-parameter semigroups of unitary endomorphisms of standard subspaces

at the Lie-Group seminar of the Friedrich-Alexander-University Erlangen-Nürnberg, Germany

Abstract: Starting with a von Neumann algebra that has a cyclic and separating vector, the Tomita-Takesaki theory allows to translate this setting to the theory of standard subspaces. Here, motivated by the Haag-Kastler theory of local observables in Quantum Field Theory, one considers certain groups of unitary operators and is interested in the semigroup of these unitaries that map the standard subspace into itself. Classical results in this context are given in the works of Borchers and Wiesbrock and by Longo/Witten.
In my PhD project I primarily focus on one-parameter semigroups of unitary endomorphisms of standard subspaces. I show a way how to link these to reflection positive Hilbert spaces and Hankel operators and in this context I investigate the unitary one-parameter semigroups appearing in the works by Longo and Witten that come from so called Pick functions.

2021

Reflection positivity and unimodular symbols for positive Hankel operators

at the Lie-Group seminar of the Friedrich-Alexander-University Erlangen-Nürnberg, Germany

Pick functions and an application to Hankel operators on Hardy spaces

at the Lie-Group seminar of the Friedrich-Alexander-University Erlangen-Nürnberg, Germany