Title: The excitation spectrum of two-dimensional Bose gases in the Gross-Pitaevskii regime
Abstract: I will discuss spectral properties of two dimensional Bose gases confined in a unit box with periodic boundary conditions. We assume that N particles interact through a repulsive two-body potential, with scattering length that is exponentially small in N, i.e. the Gross-Pitaevskii regime.
We proved that bosons in this regime exhibit complete Bose-Einstein condensation and we established the validity of the prediction of Bogoliubov theory. In particular, we determined the ground state energy expansion of the Hamilton operator up to second order correction, and the low-energy excitation spectrum.
This is a joint work with Serena Cenatiempo and Benjamin Schlein.
Title: A Lieb-Thirring Inequality For Extended Anyons
Abstract: We derive a Pauli exclusion principle for extended fermion-based anyons of any positive radius and any non-trivial statistics parameter. We consider N 2D fermionic particles coupled to magnetic flux tubes of non-zero radius, and prove a Lieb-Thirring inequality for the corresponding many-body kinetic energy operator. The implied constant is independent of the radius of the flux tubes, and proportional to the statistics parameter.
16.11.2022: Simone Rademacher (LMU Munich).
Title: The effective mass problem for the Landau-Pekar equations
Abstract: We provide a definition of the effective mass for the classical polaron described by
the Landau–Pekar equations. It is based on a novel variational principle, minimizing the energy functional
over states with given (initial) velocity. The resulting formula for the polaron's effective mass agrees with
the prediction by Landau and Pekar. This is joint work with Dario Feliciangeli and Robert Seiringer
23.11.2022: Tobias Ried (MPI MiS Leipzig).
Title: A variational approach to the regularity of optimal transportation
Abstract: In this talk I want to present a purely variational approach to the regularity theory for
the Monge-Ampère equation, or rather optimal transportation, introduced by Goldman—Otto. Following De Giorgi’s strategy
for the regularity theory of minimal surfaces, it is based on the approximation of the displacement by a harmonic gradient,
which leads to a one-step improvement lemma, and feeds into a Campanato iteration on the C^{1,\alpha}-level
for the displacement.
The variational approach is flexible enough to cover general cost functions by importing the concept
of almost-minimality: if the cost is quantitatively close to the Euclidean cost function |x-y|^2,
a minimiser for the optimal transport problem with general cost is an almost-minimiser for the one with quadratic cost.
This allows us to reprove the C^{1,\alpha}-regularity result of De Philippis—Figalli, while bypassing Caffarelli’s
celebrated theory. (This is joint work with F. Otto and M. Prod’homme)
30.11.2022: Lea Boßmann (LMU Munich).
Title: Edgeworth expansion for the weakly interacting Bose gas
Abstract: We consider the ground state and the low-energy excited states of a system of weakly interacting bosons.
We derive an Edgeworth expansion for the fluctuations of bounded one-body operators around the condensate,
which yields corrections to a central limit theorem. Based on joint work with Sören Petrat.
14.12.2022: Sabiha Tokus (Uni. Tübingen).
Title: Self-adjoint extensions of gapped operators
Abstract: Whereas many typical operators we encounter in non-relativistic quantum mechanics are bounded
from below and thus admit a self-adjoint extension according to Friedrichs’ construction,
this is not true for a generic Dirac-type operator which does not have a lower bound.
This raises the question of finding a self-adjoint extension for classes of symmetric operators that are not semi-bounded.
With inspiration from work by Esteban, Dolbeault, Loss and Séré, we find a class of operators, satisfying a gap condition,
for which we can construct a distinguished self-adjoint extension and a corresponding min—max principle,
similarly as in the Friedrichs extension. This is joint work with Lukas Schimmer and Jan Philip Solovej.
18.01.2023: Sören Petrat (Jacobs University).
Title: The Binding Energy in the Weakly Interacting Bose Gas Beyond Bogoliubov Theory
Abstract: I will give a summary of recent results on asymptotic expansions for non-relativistic
bosons in the mean-field limit. These expansions are around Bogoliubov theory and provide approximations
for the Bose gas to any order in the small parameter N^(-1/2), where N is the number of particles.
The expansions were proven for both the dynamics, and the low-lying energies and eigenstates for particles
with pair interaction. We have proven an additional result for the dynamics of bosons that are coupled
linearly to a Nelson-type quantum field. The talk will in particular emphasize the explicit nature of the expansion,
i.e., how the expansion can be used to compute physical quantities such as the binding energy,
in a way that is independent of the particle number N. This is joint work with Lea Bossmann, Marco Falconi,
Nikolai Leopold, David Mitrouskas, Peter Pickl, Robert Seiringer, and Avy Soffer.
08.02.2023: August Bjerg (University of Copenhaguen).
Title: On the scattering lengths of finite and infinite Thomas-Fermi atomsfie
Abstract: After an introduction to the scattering length we discuss the Schrödinger operators associated to finite and infinite Thomas-Fermi atoms. The focus will be on the 1-dimensional problem. In the finite case the scattering length will be simply that of minus the Thomas-Fermi potential, whereas for the infinite atoms we define it through a choice of self-adjoint extension of the associated Schrödinger operator. The key take-away is that if we choose a sequence of finite atoms with increasing atomic number but such that their scattering length is fixed then the associated operators converge in the strong resolvent sense towards that of the infinite atom with the corresponding scattering length. If time permits we might have a look at the asymptotic behavior of the scattering lengths of the Thomas-Fermi potentials or briefly discuss the (physically more relevant) radially symmetric 3-dimensional analogue of the problems treated. Based on work in progress with Jan Philip Solovej.
