GAuS Seminar

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.

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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.

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.

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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.

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