Analysis and Operator Theory Seminar

Analysis and Operator Theory Seminar
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The Ohio State University: College of Arts and Sciences
The organizers of the seminar for 2025-2026 are
Ovidiu Costin
Jan Lang
, and
Jonathan Stanfill
Date/Time
Location
Speaker
Institution
Title (click to see abstract)
Slides
September 25 at 12:40 p.m.
MW 154
Michał Wojciechowski
Institute of Mathematics of the Polish Academy of Sciences
On the Mityagin-DeLeeuw-Mirkhil theorem with
differential constraints
The classical result of the authors mentioned in the title
says that the uniform norm of Q(D)f is bounded by the uniform
norms of Q_j(D)f, j=1,2,…,k for functions f of compact support
if and only if Q is in the span of Q_j’s (here Q and Q_j’s are
homogeneous polynomials of the same degree). In the talk
I will present the analogs of this result with additional constraint
that the estimate holds only for f being a solution of differential
equation P(D)f=0.
This is joint work with Eduard Curca
October 9 at 12:40 p.m.
MW 154
Christoph Fischbacher
Baylor University
Non-selfadjoint operators with non-local point interactions
In this talk, I will discuss non-selfadjoint differential operators of the form
$i\frac{d}{dx}+V+k\langle \delta,\cdot\rangle$ and $-\frac{d^2}{dx^2}+V+k\langle \delta,\cdot\rangle$, where $V$ is a bounded complex potential. The additional term, formally given by $k\langle \delta,\cdot\rangle$, is referred to as “non-local point interaction” and has been studied in the selfadjoint context by Albeverio, Cojuhari, Debowska, I.L. Nizhnik, and L.P. Nizhnik.
I will begin with a discussion of the spectrum of the first-order operators on the interval and give precise estimates on the location of the eigenvalues. Moreover, we will show that the root vectors of these operators form a Riesz basis. If the initial operator is dissipative (all eigenvalues have nonnegative imaginary part), I will discuss the possibility of choosing the non-local point interaction in such a way that it generates a real eigenvalue even if the potential is very dissipative.
After this, I will focus on the dissipative second order-case and show similar results on constructing realizations with a real eigenvalue.
Based on previous and ongoing collaborations with Matthias Hofmann, Andrés Lopez Patiño, Sergey Naboko, Danie Paraiso, Chloe Povey-Rowe, Monika Winklmeier, Ian Wood, and Brady Zimmerman.
November 6 at 1:30 p.m
(change of time)
Journalism Building
JR 295
(change of place)
Pavel Zatitskii
University of Cincinnati
Extremal problems and monotone rearrangement on averaging classes
We will discuss integral extremal problems on the so-called averaging classes of functions, meaning classes defined in terms of averages of their elements, such as BMO, VMO, and Muckenhoupt weights. A typical extremal problem we consider involves an integral inequality, such as the John–Nirenberg inequality for BMO. One common way to formulate such questions is using Bellman functions. It turns out that such Bellman functions are solutions to specific boundary value problems, formulated in terms of convex geometry. We will also discuss the monotone rearrangement operator acting on the averaging classes, which arises naturally in this context and is useful when solving extremal problems.
November 20 at 12:40pm
MW 154
Scott Zimmerman
Ohio State
Bi-Lipschitz segments in metric spaces
A bi-Lipschitz segment in a metric space $X$ is the image of an interval in the real line under a bi-Lipschitz map. A natural question is as follows: when is a subset of a metric space contained in a bi-Lipschitz segment? In other words, given a set $K \subset \mathbb{R}$ and a bi-Lipschitz map $f:K \to X$, when is there a bi-Lipschitz extension $F:I \to X$ where $I$ is an interval containing $K$? This question was answered in the case $X = \mathbb{R}^n$ by David and Semmes for $n \geq 3$ and later by MacManus when $n = 2$. David and Semmes originally proved this result as part of their celebrated work in quantitative rectifiability. In this talk, I will discuss a recent preprint in which we prove this bi-Lipschitz extension result in a general setting when $X$ is one of a large class of metric spaces possessing certain geometric properties (namely Ahlfors regularity and supporting a Poincar\’{e} Inequality). This is joint work with Jacob Honeycutt and Vyron Vellis.
December 4 at 12:40 p.m.
MW 154
Aurel Stan
Ohio State
Necessary conditions for a linear operator to be number operator.
TBD
Decenber 11 at 12:40 p.m.
MW 154
Marjie Drake
Ohio State
TBD
TBD
Januarly 15 at 12:40 p.m.
MW 154
Michael Roysdon
University of Cincinnati
Title: Equivalence between Brunn-Minkowski type inequalities and their functional counterparts.
Feb. 19th at 12:40 p.m.
MW 154
Michael Penrod
University of Cincinnati
TBD
March 5 at 12:40 p.m.
MW 154
Chris Marx
Oberlin College
Ergodic Schrödinger Operators on the Bethe Lattice and a Modified Thouless Formula
Random Schrödinger operators on the Bethe lattice received considerable attention in the literature as an interesting `intermediary’ between one-dimensional and multidimensional results. While the tree structure of the Bethe lattice gives rise to a recursive structure similar to one-dimensional Schrödinger operators, the geometry of the Bethe lattice implies exponential growth of the surface-volume ratio, which is in stark contrast to Schrödinger operators on $Z^d$. This feature presents difficulties when trying to approximate infinite volume quantities, e.g. spectral averages for functions of the infinite volume Hamiltonian, by finite volume restrictions.

In this talk, we will explain the basic set-up for general ergodic Schrödinger operators on the Bethe lattice and present results on the limiting behavior of finite volume restrictions of functions of the Hamiltonian. We will use these results to present a generalization of Thouless’ formula to ergodic Schrödinger operators on the Bethe lattice. The Thouless formula connects the Lyapunov exponent (defined as the exponential decay rate of the Green function) to the density of states.The talk is based on joint work with Peter D. Hislop.
March 26th at 12:40 p.m.
MW 154
Bart Rosenzweig
The Ohio State University
Maximally dissipative and self-adjoint extensions of $K$-invariant operators
Let $S$ be a nonnegative symmetric operator on a Hilbert space, and let $K$ be a bounded and boundedly invertible operator. We say $S$ is $K$-invariant if it satisfies $K^*SK=S$. In this talk we give conditions for self-adjoint and maximally dissipative extensions of $S$ to preserve $K$-invariance. For example, the Friedrichs and Krein-von Neumann extensions are shown to always be $K$-invariant. We apply our results to the case of Sturm-Liouville operators where $K$ is given by $Kf(x) = A(x)f(\phi(x))$ under appropriate assumptions. Sufficient conditions on the coefficient functions in the differential operator for $K$-invariance to hold are shown to be related to Schröder’s equation, and all $K$-invariant self-adjoint extensions are then characterized. Explicit examples are discussed including a Bessel-type Schrödinger operator satisfying a nontrivial $K$-invariance on the half-line. This is based on joint work with Christoph Fischbacher and Jonathan Stanfill.
April 16th at 12:40 p.m.
MW 154
Efstathios K Chrontsios Garitsis
The University of Tennessee, Knoxville
TBD
April 23rd at 12:40pm
MW 154
Jan Lang
OSU
Degrees of non-compactness for Fourier Transform
TBD
* Joint
PDE Seminar
talk
† Joint
Harmonic Analysis and Several Complex Variables Seminar
talk