**G**erman

**Au**strian

**S**wiss Seminar on Analysis and PDE

The aim of this seminar is to stimulate scientific exchange between
the fields of analysis, partial differential equations, and
mathematical physics in a casual atmosphere. We are looking forward
to talks by young as well as senior researchers.

The seminar takes place monthly. We meet on **every first Monday of a
month at 5-6 pm CE(S)T**.
(Convert CE(S)T to your local time here.)
For the precise dates and possible exceptions, please see the table
below. We hope to see you there!

If you are interested to participate, please **subscribe to the
mailing list ^{(1)} or
contact one of the organizers:**

Konstantin Merz (TU Braunschweig, Germany)

Simone Rademacher (LMU Munich, Germany)

Christoph Kehle (ETH-ITS Zurich, Switzerland)

For an abstract of the talk, click on its title or see below.

__Seminars October 2023 - July 2024__

Date | Speaker | Title | Notes |
---|---|---|---|

November 20, 2023 | Xiaoyan Su | The \(W^{s,p}\)-boundedness of wave operators for the Schrödinger operator with inverse-square potential | Postponed from November 06. |

December 11, 2023 | Matilde Gianocca | Morse Index Stability for Conformally Invariant Lagrangians in two dimensions | Moved from December 04. |

January 9, 2024 | Michał Kijacko | Properties of weighted fractional Sobolev spaces | Moved from January 08. |

February 12, 2024 | Nikolai Leopold | Derivation of the Vlasov-Maxwell system from the Maxwell-Schrödinger equations with extended charges | Moved from February 05. |

March 04, 2024 | TBA | TBA | |

April 08, 2024 | TBA | TBA | |

May 06, 2024 | TBA | TBA | |

June 03, 2024 | TBA | TBA | |

July 01, 2024 | TBA | TBA |

** Xiaoyan Su (Loughborough University): The \(W^{s,p}\)-boundedness of wave operators for the Schrödinger operator with inverse-square potential **

In this talk, we focus on the Schrödinger operator with inverse-square potential
\(\mathcal L_a=-\Delta+\tfrac{a}{|x|^2}\), \(a\geq -\tfrac{(d-2)^2}{4}\), \(d\geq 2\).
We will discuss the boundedness of wave operators in certain Sobolev spaces,
which lead to a series of interesting inequalities, such as dispersive estimates,
Strichartz estimates and uniform Sobolev inequalities.
We will explain how to construct the wave operators using Mellin transform and
spherical harmonic decomposition, and prove that they are \(W^{s,p}\)-bounded for
certain \(p\) and \(s\) which depend on \(a\).
This talk is based on joint work with Changxing Miao and Jiqiang Zheng.

** Matilde Gianocca (ETH Zürich): Morse Index Stability for Conformally Invariant Lagrangians in two dimensions **

In the first part of the talk we will give an introductory explanation of the
so-called bubble-tree convergence for critical points of conformally invariant
Lagrangians on Riemann Surfaces. We will then proceed to discuss
Morse Index Stability along these sequences, which relies on
\(L^{2,1}\)-quantization results.
The talk is based on joint work with F. Da Lio and T. Rivière.

** Michał Kijacko (Wrocław University of Science and Technology): Properties of weighted fractional Sobolev spaces **

The talk is devoted to the properties of fractional Sobolev spaces equipped
with weights being powers of the distance to the domain. We present results
concerning density of smooth, compactly supported functions in such spaces
on bounded domains and comparability between full and truncated weighted
Gagliardo seminorms. In addition, we discuss recently obtained weighted
fractional Hardy inequalities with sharp constants and asymptotic formulae
for weighted Gagliardo seminorms.

** Nikolai Leopold (Universität Basel): Derivation of the Vlasov-Maxwell system from the Maxwell-Schrödinger equations with extended charges **

In this talk I will consider the Maxwell-Schrödinger equations in the
Coulomb gauge describing the interaction of extended fermions with their
self-generated electromagnetic field. In the first part, I will explain how
these equations heuristically emerge from non-relativistic quantum
electrodynamics in a mean-field limit of many fermions.
The second part is dedicated to the rigorous derivation of the Vlasov-Maxwell
dynamics from the Maxwell-Schrödinger equations with extended charges
in the semiclassical regime.
The talk is based on a joint work with Chiara Saffirio.

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__Seminars October 2022 - July 2023__

__Seminars October 2022 - July 2023__

** Julien Ricaud (École Polytechnique): Spectral Stability in the nonlinear Dirac equation with Soler-type nonlinearity **

This talk concerns the (generalized) Soler model: a nonlinear (massive)
Dirac equation with a nonlinearity taking the form of a space-dependent
mass. The equation admits standing wave solutions and they are generally
expected to be stable (i.e., small perturbations in the initial conditions
stay small) based on numerical simulations. However, contrarily to the
nonlinear Schrödinger equation for example, there are very few
results in this direction.
The results that I will discuss concern the simpler question of spectral
stability (and instability), i.e., the absence (or presence) of
exponentially growing solutions to the linearized equation around a
solitary wave. As in the case of the nonlinear Schrödinger equation,
this is equivalent to the presence or absence of "unstable eigenvalues"
of a non-self-adjoint operator with a particular block structure.
I will highlight the differences and similarities with the
Schrödinger case, present some partial results for the
one-dimensional case, and discuss open problems.

This is joint work with Danko Aldunate, Edgardo Stockmeyer and
Hanne van den Bosch.

** Barbara Roos (Institute of Science and Technology Austria): Boundary superconductivity in the BCS model **

We consider the linear BCS equation, determining the BCS critical
temperature, in the presence of a boundary, where Dirichlet boundary
conditions are imposed. In the one-dimensional case with point interactions,
we prove that the critical temperature is strictly larger than the bulk
value, at least at weak coupling. We also show that the relative shift
in critical temperature vanishes if the coupling constant either goes to
zero or to infinity.

** Rodrigo Matos (Texas A&M University): Irreducibility of the Bloch and Fermi varieties on periodic media and connections to spectral theory **

The structure of the dispersion relation is one of the central aspects
to the study of periodic Schroedinger operators. Besides the intrinsic
interest from the viewpoint of several complex variables, the dispersion
relation also carries relevant information for the spectral theory of
periodic media. In particular, for the structure of spectral boundaries,
isospectrality, and existence of eigenvalues for locally perturbed
operators. I will discuss some of these connections as well as recent
irreducibility theorems for the Bloch and Fermi varieties, focusing on the
joint work with Jake Fillman and Wencai Liu
(arXiv:2107.06447, J. Funct. Anal. 2022).
This recent paper covers a wide class of lattice geometries in arbitrary
dimension and verifies the discrete version of a conjecture of Kuchment
for various models.
Time allowing, I will also comment on future directions and ongoing work.

** TBA (TBA): TBA **

TBA

** Yukimi Goto (Kyushu University): Spontaneous mass generation and chiral symmetry breaking in a lattice Nambu-Jona-Lasinio model (arXiv:2209.06031) **

In quantum chromodynamics, without interactions, quarks have no mass and
a conserved quantity called chirality. In reality, quarks have mass and
chiral symmetry is broken. This is thought to be the result of spontaneous
symmetry breaking by the interaction. In this talk we consider a lattice
version of the Nambu-Jona-Lasinio model with interacting staggered fermions
in the Kogut-Susskind Hamiltonian formalism. In a strong coupling regime
for the four-fermion interaction, we prove that the mass of the fermions is
spontaneously generated at sufficiently low temperatures. This implies that
the chiral symmetry is spontaneously broken if the continuum limit exists.
For the proof, the reflection positivity for fermions and the infrared bound
method are crucial. This talk is based on joint work with Tohru Koma.

** Marios Antonios Apetroaie (University of Toronto): Instability of gravitational and electromagnetic perturbations of extremal Reissner-Nordström spacetime **

Gravitational and electromagnetic perturbations for the full subextremal range,
\(|Q| < M \), of Reissner-Nordström spacetimes, as solutions to the
Einstein-Maxwell equations, have been shown to be linearly stable. We address
the aforementioned problem for the extremal, \(|Q|=M\), Reissner-Nordström
spacetime, and contrary to the subextremal case we see that instability
results hold, manifesting along the future event horizon of the black hole,
\(H^+\). In particular, depending on the number of translation invariant
derivatives of derived gauge-invariant quantities, we show decay, non-decay,
and polynomial blow-up estimates asymptotically along \(H^+\).
As a consequence, we show that solutions to the generalized Teukolsky system
of positive and negative spin satisfy analogous estimates. Stronger and
unprecedented instabilities are realized for the negative spin solutions,
with the extreme curvature component, \(\underline{a}\), not decaying
asymptotically along the event horizon.

** Jinyeop Lee (Ludwig-Maximilians-Universität München): On the characterisation of fragmented Bose-Einstein condensation and its emergent effective evolution **

Fragmented Bose-Einstein condensates are large systems of identical bosons
displaying multiple macroscopic occupations of one-body states, in a suitable
sense. The quest for an effective dynamics of the fragmented condensate at
the leading order in the number of particles, in analogy to the much more
controlled scenario for complete condensation in one single state, is
deceptive both because characterising fragmentation solely in terms of reduced
density matrices is unsatisfactory and ambiguous, and because as soon as the time
evolution starts the rank of the reduced marginals generically passes from
finite to infinite, which is a signature of a transfer of occupations on
infinitely many more one-body states. In this work we review these difficulties,
we refine previous characterisations of fragmented condensates in terms of
marginals, and we provide a quantitative rate of convergence to the leading
effective dynamics in the double limit of infinitely many particles and
infinite energy gap.
This is a joint work with Alessandro Michelangeli.

** Kouichi Taira (Ritsumeikan University, Kyoto): Essential self-adjointness of Klein-Gordon operators on asymptotically Minkowski spacetime **

In this talk, I will talk about essential self-adjointness (ESS) of Klein-Gordon
operators on spacetimes which are perturbations of Minkowski spacetime.
ESS of differential operators has been studied especially for elliptic operators
such as Laplace-Beltrami operators and Schrödinger operators.
Here we focus on ESS for Klein-Gordon operators which are not elliptic,
where previous methods cannot be applied. Moreover, it has an application to a
construction of Feynman propagator in Quantum Field Theory.
This is a joint work with Shu Nakamura.

** Haruya Mizutani (Osaka University): Strichartz estimates for Schrödinger equations with long-range potentials **

The Strichartz estimate is one of fundamental tools in the study of nonlinear
dispersive equations. This talk deals with (global-in-time) Strichartz estimates
for Schrödinger equations with potentials decaying at infinity.
The case when the potential decays sufficiently fast has been extensively studied
in the last three decades. However, it has remained mostly unknown for slowly
decaying potentials in which case the standard perturbation argument does not work.
We instead employ several techniques from long-range scattering theory and
microlocal/semiclassical analysis, and prove Strichartz estimates for a class of
positive potentials decaying arbitrarily slowly. A typical example is the positive
Coulomb potential in three space dimensions.
As an application, we also obtain a modified scattering type result for the final
state problem of the nonlinear Schrödinger equations with long-range
nonlinearity and potential.
This is partly joint work with Masaki Kawamoto (Ehime University).

** Satoshi Masaki (Hokkaido University): On the standing-wave solutions to standard forms of a class of nonlinear Schrödinger systems **

In this talk, we consider 2-coupled systems of nonlinear Schrödinger
equations with cubic nonlinearities. One typical system in this class is the
Manakov model. In the first part, we consider the classification of 2-coupled
systems. The complete list of standard forms of systems with Hamiltonian
structure is given. This is due to a representation of a system in terms of a
pair of a matrix and a vector which clarifies the Hamiltonian structure of
the systems. In the second part, we consider standing-wave solutions to the
standard forms. To this end, we introduce an abstract treatment that is
applicable to a wider class of \(N\)-coupled systems. It turns out that the
existence/nonexistence and the shape of the ground state are well-described
by the function determining the nonlinear potential part.

__Seminars October 2021 - July 2022__

__Seminars October 2021 - July 2022__

** Peer Kunstmann (Karlsruhe Institute of Technology): On the NLS outside the usual settings **

We present results on local and global wellposedness of the one-dimensional
Schrödinger equation on the real line, mostly for the cubic case.
Partly they rely on a normal form type reduction via the differentiation
by parts technique that has been introduced for periodic problems.
We show that this technique can be modified to work in certain modulation
spaces \(M^s_{p,q}(\mathbb{R})\). Moreover, we explain how this idea can
be used in a hybrid problem for non-decaying and non-periodic data in
\(H^{s_1}(\mathbb{R})+H^{s_2}_\mathrm{per}(\mathbb{R})\) where
\(0\le s_1\le s_2\).
Finally we discuss the role of "energies" that can be used to obtain global
results for such hybrid problems.

This reports on joint work with L. Chaichenets, D. Hundertmark, F. Klaus, and N. Pattakos.

** Jacob Shapiro (Princeton University): Delocalization in the integer-valued Gaussian Field and the BKT phase of the 2D Villain model **

It is shown that the Villain model of two-component spins over two
dimensional lattices exhibits slow, non-summable, decay of correlations at
any temperature at which the dual integer-valued Gaussian field exhibits
delocalization. For the latter, we extend the recent proof by Lammers of a
delocalization transition in two dimensional graphs of degree three, to all
doubly periodic graphs, in particular to \(\mathbb{Z}^2\). Taken together
these two statements yield a new perspective on the BKT phase transition in
the Villain model, and a new proof on delocalization in two dimensional
integer-valued height functions.
Joint with Aizenman, Harel and Peled.

** Giada Franz (ETH Zürich): Equivariant min-max theory to construct free boundary minimal surfaces in the unit ball **

I will start motivating the study of free boundary minimal surfaces (FBMS)
in the three-dimensional Euclidean unit ball, namely critical points of the
area functional with respect to variations that constrain their boundary to
the boundary of the ball (i.e., the unit sphere). Then, I will present an
equivariant version of Almgren-Pitts min-max theory, which turns out to be
a powerful tool to construct and investigate FBMS in the unit ball with a
given topology.

** Michael Hott (The University of Texas at Austin): On the emergence of a quantum Boltzmann equation in the presence of a Bose-Einstein condensate **

The mathematically rigorous derivation of a nonlinear Boltzmann equation
from first principles is an extremely active research area. In classical
physical systems, this has been achieved in various models, based on a
variety of fundamental works. In the quantum case, the problem has
essentially remained open. I will explain how a cubic quantum Boltzmann
equation arises within the fluctuation dynamics of a Bose-Einstein
condensate, starting with the von Neumann equation for an interacting Boson
gas.
This is based on joint work with Thomas Chen.

** Charlotte Dietze (Ludwig-Maximilians-Universität München): Dispersive Estimates for Nonlinear Schrödinger Equations with External Potentials **

We consider the long time dynamics of nonlinear Schrödinger equations
with an external potential. More precisely, we look at Hartree type
equations in three or higher dimensions with small initial data. We prove
an optimal decay estimate, which is comparable to the decay of free
solutions. Our proof relies on good control on a high Sobolev norm of the
solution to estimate the terms in Duhamel's formula.

** Dejan Gajic (Radboud University): Late-time tails for geometric wave equations with inverse-square potentials **

I will introduce a new method for obtaining the precise late-time
asymptotic profile of solutions to geometric wave equations with
inverse-square potentials on asymptotically flat spacetimes. This setting
serves as a convenient toy model for understanding novel dynamical
properties in the context of Einstein's equations of general relativity
that arise in a variety of situations, e.g. when considering the
gravitational properties of electromagnetically charged matter, when
describing dynamical, rapidly rotating black holes and when considering
higher, odd, spacetime dimensions.

** Fabio Pizzichillo (Universidad de Cantabria): Boundary value problems for 2-D Dirac operator on corner domains and the Coulomb interaction **

This talk aims to present results on the self-adjoint extensions of
Dirac operators on plane domains with corners in dimension two.
We consider the case of infinite-mass boundary conditions and we obtain
explicitly the self-adjoint extensions of the operator.
It turns out that the presence of corners typically spoils the elliptic
regularity known to hold for smooth boundaries.
Then we discuss the self-adjointness and some spectral properties of
these operators in presence of a Coulomb-type potential with the
singularity placed on the vertex.
This is a collaboration work with Hanne Van Den Bosch, Biagio Cassano
and Matteo Gallone.

** TBA (TBA): TBA**

TBA

** Fatima-Ezzahra Jabiri (University College London): On the stability of trapped timelike geodesics in non-vacuum black hole spacetime **

The center of most galaxies can be described by a black hole with matter
orbiting around it. In the context of relativistic kinetic theory, the
Vlasov matter model is used to describe the center of galaxies, where the
stars play the role of gas particles and collisions among them are
neglected so that gravity is the only interaction taken into account.
In this setting, stars are assumed to move along timelike future directed
geodesics in a given spacetime. In this talk, we shall be interested in
the final states of such self-gravitating systems. These can be described
by stationary black hole solutions to the so-called Einstein-Vlasov system.
More precisely, I will show a stability result for trapped timelike
geodesics and discuss the ideas behind the construction of these final
states.

** Emanuela Giacomelli (Ludwig-Maximilians Universität München): On the Huang-Yang order correction for the dilute Fermi gas **

We consider \(N\) spin \(1/2\) fermions interacting with a positive and
regular enough potential in three dimensions. We compute the ground state
energy of the system in the dilute regime making use of the almost-bosonic
nature of the low-energy excitations of the systems.

__Seminars November 2020 - July 2021__

__Seminars November 2020 - July 2021__

** Elena Giorgi (Princeton University): The stability of charged black holes **

I will start by motivating the study of black holes and
introducing the problem of their stability as solutions to the Einstein
equation. I will then concentrate on the case of charged black holes and
their interaction with electromagnetism. From the prospective of PDEs, I
will especially focus on two aspects of the resolution of the problem:
the identification of gauge-invariant quantities, and the analysis of
coupled systems of wave equations.

** Christian Brennecke (Harvard University): Bose-Einstein Condensation beyond the Gross-Pitaevskii Regime **

In this talk, I will consider Bose gases in a box of volume one that
interact through a two-body potential with scattering length of the order
\(N^{-1+\kappa}\), for \(\kappa >0\). For small enough
\(\kappa \in (0;1/43)\), slightly beyond the Gross-Pitaevskii regime
(\(\kappa=0\)), I will explain a proof of Bose-Einstein condensation for
low-energy states that provides bounds on the expectation and on higher
moments of the number of excitations.
The talk is based on joint work with A. Adhikari and B. Schlein.

** Lucrezia Cossetti (Karlsruhe Institute of Technology): Absence of eigenvalues of Schrödinger, Dirac and Pauli Hamiltonians via the method of multipliers **

Originally arisen to understand characterizing properties connected with
dispersive phenomena, in the last decades the method of multipliers has
been recognized as a useful tool in Spectral Theory, in particular in
connection with proof of absence of point spectrum for both self-adjoint
and non self-adjoint operators. In this seminar we will see
the developments of the method reviewing some recent results concerning
self-adjoint and non self-adjoint Schrödinger operators in different
settings, specifically both when the configuration space is the whole
Euclidean space \(\mathbb{R}^d\) and when we restrict to domains with
boundaries. We will show how this technique allows to detect physically
natural repulsive and smallness conditions on the potentials which
guarantee total absence of eigenvalues. Some very recent results
concerning Pauli and Dirac operators will be presented as well. The talk
is based on joint works with L. Fanelli and D. Krejcirik.

** John R. L. Anderson (Princeton University): Stability results for anisotropic systems of wave equations **

In this talk, I will describe a global stability result for a nonlinear
anisotropic system of wave equations. This is motivated by studying
phenomena involving characteristics with multiple sheets. For the proof, I
will describe a strategy for controlling the solution based on bilinear
energy estimates. Through a duality argument, this will allow us to prove
decay in physical space using decay estimates for the homogeneous wave
equation as a black box. The final proof will also require us to exploit a
certain null condition that is present when the anisotropic system of wave
equations satisfies a structural property involving the light cones of the
equations.

** Sabine Boegli (Durham University): On Lieb-Thirring inequalities for one-dimensional non-self-adjoint Jacobi and Schroedinger operators **

In the first part of this talk I will give an introduction to spectral
theory of Schroedinger operators and their discrete analogues, the Jacobi
operators. Lieb-Thirring inequalities give estimates on the discrete
eigenvalues and their accumulation rate to the essential spectrum. Such
information is useful in quantum mechanics, but the estimates are well
understood only in the self-adjoint setting.
In the second part of the talk I will present recent results concerning
possible extensions of the Lieb-Thirring inequalities to the
non-self-adjoint setting (based on joint work with Frantisek Stampach).

** Alessandro Olgiati (Universität Zürich): Stability of the Laughlin phase in presence of interactions **

The Laughlin wave function is at the basis of the current understanding of
the fractional quantum Hall effect (FQHE) and its associated physical
features (fractional charge quasi-particles, anyonic statistics...).
Nevertheless, very few rigorous mathematical results on Laughlin theory are
available in literature.
After a brief introduction, I will present a model, within Laughlin
framework, for the response of FQHE charge carriers to variations of the
external potential and of the inter-particle interaction. Our main result
is that the energy is asymptotically captured by the minimum of an effective
functional with variational constraints fixed by the incompressibility of
the Laughlin phase. Moreover, as was already known for the non-interacting
case, we show that the one-body density converges to the characteristic
function of a set.
This is a joint work with Nicolas Rougerie (ENS Lyon).

** André Guerra (University of Oxford): Symmetry breaking and ill-posedness for the Jacobian equation **

When can a given measure be transported to a uniform measure? The answer
to this problem goes back to Von Neumann and Oxtoby-Ulam in 1941: under
natural assumptions, the transport map always exists; however, its
regularity is unclear. Although the case where the measure is sufficiently
regular is by now well understood, the low regularity problem has remained
open. In this talk I will present some of the first results in this
direction.
This is based on joint work with Lukas Koch (Oxford) and Sauli Lindberg
(Aalto).

** Mateus Costa de Sousa (Basque Center for Applied Mathematics): Interpolation formulas, sign uncertainty principles and sphere packing problems **

In this talk we will discuss how certain kinds of uncertainty principles
and interpolation formulas are connected to sphere packing problems and
talk about some recent developments on these fronts.

** Amanda Young (Technische Universität München): Spectral Gaps in Truncated Haldane Pseudopotentials **

In 1983, Haldane introduced his family of pseudopotentials as Hamiltonian
models for the fractional quantum Hall effect. While numerical works
support the conjecture that these models have a spectral gap above which
are fractional excitations, a rigorous proof has yet to be given. In this
talk, we consider the 1/2 (bosonic) and 1/3 (fermionic) pseudopotentials in
the cylinder geometry which, when written in terms of a suitable basis of
orbitals, take the form of one-dimensional quantum lattice models.
We consider effective versions of these models obtained from truncating
exponentially small (in norm) long range interaction terms, and prove the
resulting model has a nonvanishing spectral gap above the ground state.
Key to our result is our ability to identify invariant subspaces using the
concept of domino tilings of the lattice which contain the ground states
of the Hamiltonian. We also discuss how to use distorted tilings to
investigate excited states of the Hamiltonian.
This talk is based of joint work with B. Nachtergaele and S. Warzel.

[1] B. Nachtergaele, S. Warzel, and A. Young - J. Phys. A: Math. Theor. 54: 01LT01 (2020)

[2] B. Nachteorgaele, S. Warzel, and A. Young - Commun. Math. Phys. 383: 1093--1149 (2021)

[3] S. Warzel, and A. Young - in preparation

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