Oberseminar `Calculus of Variations and Applications'
Winter Term 2019/20
Date Speaker Topic Note Oct. 16, 2019 Jonas Lampart InteriorBoundary Conditions and the BogoliubovFrÃ¶hlich Hamiltonian Oct. 23, 2019 João Pedro Ramos Fourier uncertainty principles, interpolation and uniqueness sets Oct. 30, 2019 Jakob Ullmann A Generalization of Weyl's Asymptotic Formula for the Relative Trace of Singular Potentials Nov. 6, 2019 Jérémy Sok Dirac operators with magnetic links Nov. 13, 2019 Nov. 20, 2019 Niels Benedikter Correlation Energy of a Fermi Gas and Bosonization of Collective Excitations Nov. 27, 2019 Daniel Ueltschi Random loop approaches to quantum spin systems Dec. 4, 2019 Dec. 11, 2019 Carlos Perez Fractional degenerate PoincarÃ©Sobolev inequalities through Harmonic Analysis Dec. 18, 2019 Simona Rota Nodari Uniqueness and nondegeneracy for a class of semilinear elliptic equations Jan. 8, 2020 Lucrezia Cossetti Absence of eigenvalues of Schrödinger, Dirac and Pauli Hamiltonians via the method of multipliers Jan. 15, 2020 Jan. 22, 2020 Jan. 29, 2020 Tiago Picon HardyLittlewoodSobolev inequalities for homogeneous differential operators Feb. 5, 2020
Abstracts

 Jonas Lampart (LMU): InteriorBoundary Conditions and the BogoliubovFrÃ¶hlich Hamiltonian
 The BogoliubovFrÃ¶hlich Hamiltonian models the interaction of an impurity particle with excitations of a dilute BoseEinstein condensate. Because of the dilute nature of the system, the interactions can be modelled as contact interactions. However, this introduces a strong ultraviolet singularity into the mathematical formulation. I will discuss how to deal with these singularities using interiorboundary conditions. This approach can be applied to general ultraviolet singularities in (nonrelativistic) quantum field theories, and aims to describe the domain of the Hamiltonian by generalised boundary conditions relating wavefunctions corresponding to different particle numbers. I will also explain the relation of this approach to renormalisation.

 João Pedro Ramos (Bonn): Fourier uncertainty principles, interpolation and uniqueness sets
 A classical result in the theory of entire functions of exponential type, Shannon?s interpolation formula predicates that, given a function whose Fourier transform vanishes outside the interval $[1/2,1/2]$, it is possible to recover it from its values at the integers. More specifically, it holds, in a suitable sense of convergence, that $$ f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(xn))}{\pi(x n)}. $$ This formula is unfortunately unavailable for arbitrary Schwartz functions on the real line, but a recent result of Radchenko and Viazovska provides us with an explicit construction of an interpolation basis for even Schwartz functions. It states, in a nutshell, that we can recover explicitly the function given its values at the squares of roots of integers. In this expository talk, we will discuss a bit these two results, and explore, in connection to classical Fourier uncertainty results, the question of determining which pairs of sets $(A,B)$ satisfy that, if a Schwartz function $f$ vanishes on A and its Fourier transform vanishes on B, then $f \equiv 0.$

 Jakob Ullmann (LMU): A Generalization of Weyl's Asymptotic Formula for the Relative Trace of Singular Potentials
 By Weyl's asymptotic law, in $d$dimensional space, the sum of the first moments of the negative eigenvalues of a SchrÃ¶dinger operator with potential $V$ is asymptotically proportional to $h^{d}$ times the integral of $\vert V_ \vert^{1+d/2}$ in the semiclassical limit $h \downarrow 0$, whenever this integral is finite. In this thesis we show that for two SchrÃ¶dinger operators with singular potentials, theirs sums of negative eigenvalues grow faster than $h^{d}$, but if the difference of the potentials is small enough and the integral over $\vert V_{1} \vert^{1+d/2}  \vert V_{2} \vert^{1+d/2}$ is finite, the difference of their sums of negative eigenvalues is still asymptotically of order $h^{d}$.

 Jérémy Sok (Basel): Dirac operators with magnetic links
 We investigate the zero modes for threedimensional Dirac operators with singular magnetic fields supported on links. They can be seen as a generalization of AharonovBohm solenoids, in particular they exhibit the same $2\pi$periodicity of the fluxes carried by the field lines. The occurrence of zero modes is studied through the spectral flow of loops of such singular operators: it is generically nonzero and depends on the geometry of the field lines (not only their topology). This a joint work with Fabian Portmann and Jan Philip Solovej.

 Niels Benedikter (Klosterneuburg): Correlation Energy of a Fermi Gas and Bosonization of Collective Excitations
 While bosonization is a rigorously established and exact method for analyzing onedimensional systems, it is difficult to generalize to higher dimensional systems. I am going to discuss how bosonization can be generalized as an approximation for the threedimensional Fermi gas at high density. In particular I am going to show how bosonization can be used to systematically derive an upper bound on the correlation energy of a Fermi gas. I am also going to comment on further physical predictions obtainable from the bosonized theory.

 Daniel Ueltschi (Warwick): Random loop approaches to quantum spin systems
 Quantum spin systems are used to describe the magnetic properties of condensed matter physics. I will introduce a family of models that includes the ferromagnetic and antiferromagnetic Heisenberg models, and describe their random loop representations. The joint distribution of loop lengths is conjectured to be a PoissonDirichlet distribution and it is related to the nature of symmetry breaking. This is expected to hold for lattices of dimensions 3 and higher, but this can only be proved on the complete graph. (Joint work with J. BjÃ¶rnberg and J. FrÃ¶hlich.)

 Carlos Perez (Bilbao): Fractional degenerate PoincarÃ©Sobolev inequalities through Harmonic Analysis
 In this lecture we will discuss some recent results concerning fractional PoincarÃ© and PoincarÃ©Sobolev inequalities with weights. These results improve some classical estimates due to FabesKenigSerapioni obtained in the 80?s in connection with the local regularity of solutions of appropriate degenerate elliptic equations. We will show that these new general results contain as a byproduct, classical theorems like the JohnNirenberg theorem for BMO functions. Our approach is different from the usual ones which are based on representation formulae and it is based on methods from Harmonic Analysis. We will also discuss other applications of Harmonic Analysis tools to the context of PoincarÃ© inequalities. The main part of the lecture is based on a joint work with E. Rela, University of Buenos Aires.

 Simona Rota Nodari (Dijon): Uniqueness and nondegeneracy for a class of semilinear elliptic equations
 In this talk, I will present a result on the uniqueness and the nondegeneracy of positive radial solutions for a class of semilinear elliptic equations. Next, I will illustrate this result with two examples: a nonlinear SchroÌˆdinger equation for a nucleon and a SchroÌˆdinger equation with a double power nonlinearity. This talk is based on joint works with Mathieu Lewin.

 Lucrezia Cossetti (Karlsruhe): 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 selfadjoint and non selfadjoint operators. In this seminar we will see the developments of the method reviewing some recent results concerning selfadjoint and non selfadjoint SchrÃ¶dinger operators in different settings, specifically both when the configuration space is the whole Euclidean space \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 also presented. The talk is based on joint works with L. Fanelli and D. Krejcirik.

 Tiago Picon (São Paulo): HardyLittlewoodSobolev inequalities for homogeneous differential operators
 In this lecture we present a new characterization of HardyLittlewoodSobolev inequalities in $L^1$ norm for homogeneous linear differential operators with constant coefficients. Variants and applications are presented, in particular a version of $L^1$ SteinWeiss inequality for vector fields. This is based on joint work with Jorge Hounie (UFSCar  Brazil) and (working in progress) with Pablo De NÃ¡poli (UBA  Argentine).


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