In 2002, Hundertmark and Simon proved Lieb–Thirring inequalities for Jacobi operators. They conjectured that their bounds could be improved by replacing a term (which depends on the off-diagonal parts of the operator) by its positive part. In this talk, I will present a proof of their conjecture. Subsequently I will relate (sharp) Lieb–Thirring inequalities for Jacobi operators to (sharp) Lieb–Thirring inequalities for Schrödinger operators on the real line.

This talk is partly based on joint work with A. Laptev and M. Loss.

I will talk about the model of "random hyperbolic surfaces", picked according to the Weil-Petersson measure. I will be interested in the spectrum of the laplacian on these random objects, in the large volume asymptotic. I will in particular discuss the spectral gap, and will report on joint work with Laura Monk, aiming to show that the lowest eigenvalue of a random compact hyperbolic surfaces is typically greater than 1/4-ε.

We consider a system of indistinguishable classical particles in Euclidean space interacting through a short-range pair potential. Fixing the one-particle density profile of the system, we minimize the free energy over the set of states with this exact density. This can be useful e.g. to model interfaces between two different equilibrium phases of a system. In this talk, I will discuss how to approximate the free energy in terms of a local functional depending only on the density of the system. The approximation is especially useful when the density is almost constant over large regions of space. This is joint work with Mathieu Lewin and Michal Jex.

I will present recent work done with S. Kumar and F. Ponce-Vanegas and L. Roncal. We study the process of dispersion of low-regularity solutions to the free Schrödinger equation using fractional weights. We give another proof of the uncertainty principle for fractional weights and use it to get a lower bound for the concentration of mass. We consider also the evolution when the initial datum is the Dirac comb in ${\mathbb R}$. In this case we find fluctuations that concentrate at rational times and that resemble a realization of a Lévy process. Furthermore, the evolution exhibits multifractality.

In 1979, Aharonov and Casher provided a precise formula for the zero eigenspace dimension of the Pauli operator. Since then, researchers have explored various generalizations and variations of the theorem. Recently, we have studied a case in which the Neumann or Robin condition is imposed on the boundary of a bounded domain in the plane, leading to potential negative eigenvalues. Our goal is to accurately count these negative eigenvalues. While we provide an exact formula for the disc, we can only offer a lower bound for general domains. Furthermore, we examine some semi-classical implications of our findings. Joint work with Søren Fournais, Magnus Goffeng and Ayman Kachmar

We consider a gas of bosonic particles confined in a box with Neumann boundary conditions. We prove Bose-Einstein condensation in the Gross-Pitaevskii regime, with an optimal bound on the condensate depletion. Our lower bound for the ground state energy in the box implies (via Neumann bracketing) a lower bound for the ground state energy of the Bose gas in the thermodynamic limit. (Joint work with Robert Seiringer.)

I will discuss the nature of the low-temperature expansion for the multipoint spin correlations of classical O(N) vector models in dimension three and higher. The main result I will present is that such expansions define asymptotic series, with explicit bounds on the error terms associated with their finite order truncations. The result applies, in particular, to the spontaneous magnetization of the 3D Heisenberg model. The proof combines a priori bounds on the moments of the local spin observables, following from reflection positivity and the infrared bound, with an integration-by-parts method applied systematically to a suitable integral representation of the correlation functions. The method of proof generalizes an approach, proposed originally by Bricmont-Fontaine-Lebowitz-Lieb-Spencer in the context of the rotator model, to the case of non-abelian symmetry and non-gradient observables. Time permitting, I will comment on the perspectives and open problems for quantum spin systems. Joint work with Sebastien Ott.

It is well-known that the interaction between the spin and a magnetic field typically gives rise to zero energy eigenfunctions of two-dimensional Pauli and Dirac operators, the so called Aharonov-Casher states. In this talk we will consider an additional multiplicative perturbation with a small coupling parameter. We will show that if the perturbation is positive in certain weak sense, then the Aharonov-Casher states turn into resonances. We will also provide the leading terms of their lifetimes in the weak coupling expansion. The talk is based on a joint work with Jonathan Breuer (Hebrew University, Jerusalem).

The topic of this talk are quantum graphs with the vertex coupling which does not preserve the time-reversal invariance. As a case study we analyze a simple example in which the asymmetry is maximal at a fixed energy. This has an interesting consequence, namely that high-energy scattering depends crucially on the vertex parity; we will demonstrate implications of this fact for spectral and transport properties in several classes of graphs, both finite and infinite periodic ones. Furthermore, we discuss other time-asymmetric graphs and identify a class of such couplings which exhibits a nontrivial PT - symmetry despite being self-adjoint. We also illustrate how the presence or absence of the Dirichlet component in the vertex coupling is manifested in the spectrum, and finally, we demonstrate the Band-Berkolaiko universality for kagome lattices with the indicated coupling.

The results come from a common work with Marzieh Baradaran, Jiří Lipovský, and Miloš Tater.

We discuss the typical behavior of two important quantities on compact manifolds with a Riemannian metric g: the number, c(T, g), of primitive closed geodesics of length smaller than T, and the error, E(L, g), in the Weyl law for counting the number of Laplace eigenvalues that are smaller than L. For Baire generic metrics, the qualitative behavior of both of these quantities has been understood since the 1970’s and 1980’s. In terms of quantitative behavior, the only available result is due to Contreras and it says that an exponential lower bound on c(T, g) holds for g in a Baire-generic set. Until now, no upper bounds on c(T, g) or quantitative improvements on E(L, g) were known to hold for most metrics, not even for a dense set of metrics. In this talk, we will introduce the concept of predominance in the space of Riemannian metrics. This is a notion that is analogous to having full Lebesgue measure in finite dimensions, and which, in particular, implies density. We will then give stretched exponential upper bounds for c(T, g) and logarithmic improvements for E(L, g) that hold for a predominant set of metrics. This is based on joint work with J. Galkowski.

In this talk, we discuss the bulk gap for a truncated 1/3-filled Haldane pseudopotential for the fractional quantum Hall effect. For this Hamiltonian with periodic boundary conditions, we establish a spectral gap above the highly degenerate ground-state space which is uniform in the volume and particle number. These bounds are proved by identifying invariant subspaces to which we apply gap-estimate methods previously developed only for quantum spin Hamiltonians. In the case of open boundary conditions, a lower bound on the spectral gap accurately reflects the presence of edge states, which do not persist into the bulk. Customizing the gap technique to the invariant subspace, we avoid the edge states and establish a more precise estimate on the bulk gap in the case of periodic boundary conditions. The same approach can also be applied to prove a bulk gap for the analogously truncated Haldane pseudopotential with maximal half filling, which describes a strongly correlated system of spinless bosons in a cylinder geometry.

This is based off joint work with S. Warzel.

Motivated by models of anyonic systems, we investigate the self-adjointness and spectral properties of two-dimensional Schrödinger operators for one non-relativistic particle in presence of Aharonov-Bohm (AB) fluxes. We start by addressing the simple case of a single infinite realistic solenoid, i.e., with a AB flux perturbed by a regular magnetic field, and study the properties of the operator by means of quadratic form techniques. Next, we apply the obtained results to the case of many ideal fluxes on the plane. Finally, we present some work in progress about the homogenization regime where the number of fluxes goes to infinity, while the single flux intensity tends to zero in such a way that the total flux remains finite.

Joint work with Davide Fermi (Politecnico di Milano).

We study the effective dynamics of heavy, quantum-mechanical particles (tracer particles) that interact with a Bose-Einstein condensate — we assume that the tracer particle’s mass, and the size of the condensate is of the same order of magnitude, N. For large N, the dynamics of the microscopic system is expected to be described by a pair of (interacting) macroscopic variables: the classical positions of the tracer particles, and the condensate’s wave function. In this talk, we present new quantitative results concerning the validity of this approximation.

In this talk, we shall discuss how a Schrödinger equation with high dimensionality can be partitioned into weakly interacting subsystems. This type of models describes system-bath type situations where reactive molecular fragments are embedded in a large molecular bath (a protein, or a solvent). We shall consider two schemes of dimension reduction: one based on Taylor expansion (collocation) and the other one based on partial averaging (mean-field or Hartree approximation) for initial data that have a tensor-product structure. We shall also investigate the situation where one of the sets of variables is semi-classically scaled in the coupling variable. In all these cases, we shall motivate the introduction of these schemes, discuss their and we shall show numerical realization. These results are joint works with colleagues from theoretical chemistry, Irene Burghardt and Benjamin Lasorne, and from mathematics, Rémi Carles and Caroline Lasser.

Pólya’s conjecture in spectral geometry (1954) states that the eigenvalue counting functions of the Dirichlet and Neumann Laplacian on a bounded Euclidean domain can be estimated from above and below, respectively, by the leading term of Weyl’s asymptotics. Pólya’s conjecture is known to be true for domains which tile the space, and, in addition, for some special domains in higher dimensions, but is still open in other cases. I’ll present a recent joint work with Iosif Polterovich and David Sher (March 2022) in which we prove Pólya’s conjecture for the disk, making it the first non-tiling planar domain for which the conjecture is verified. Along the way, we develop the known links between the spectral problems in the disk and certain lattice counting problems using some new relations for Bessel functions. Our proofs of Pólya’s conjecture for both Dirichlet and Neumann problems are purely analytic for the values of the spectral parameter above some large (but explicitly given) number. Below that number we give rigorous computer-assisted proofs which converge in a finite number of steps and use only integer arithmetic. I will also discuss some more recent improvements obtained by Filonov (August 2022).

We consider a gas of N bosons moving on the three-dimensional unit torus and interacting through a hard sphere potential with radius of order 1/N. We prove an upper bound for the ground state energy, valid up to errors that vanish in the limit of large N.

This is joint work with G. Basti, S. Cenatiempo, A. Olgiati and G. Pasqualetti.

I will discuss some recent developments on Lieb-Thirring inequalities for the kinetic energy of orthonormal functions, focusing on direct methods initiated by Rumin and Lundholm-Solovej which do not rely on eigenvalue bounds of Schrödinger operators.

In this talk, I will present a recent result on the behaviour of the solutions of a Dirac-Klein-Gordon system in the limit of strong coupling and large masses of the Klein-Gordon fields. I will prove convergence of the solutions to the system to those of a cubic non-linear Dirac equation. This shows that in this parameter regime, which is relevant to the relativistic mean-field theory of nuclei, the retarded interaction is well approximated by an instantaneous, local self-interaction. This is a joint work with J. Lampart, L. Le Treust and J. Sabin.

In this talk I shall present constructions of Schrödinger operators with complex-valued potentials whose spectra exhibit interesting properties. One example shows that for sufficiently large $p$, namely $p'>(d+1)/2$ where $d$ is the dimension, the discrete eigenvalues need not be bounded by the $L^p$ norm of the potential. This is a counterexample to the Laptev-Safronov conjecture (Comm. Math. Phys. 2009). Another construction proves optimality (in some sense) of generalisations of Lieb-Thirring inequalities to the non-selfadjoint case - thus giving us information about the accumulation rate of the discrete eigenvalues to the essential spectrum. This talk is based on joint works with Jean-Claude Cuenin and Frantisek Stampach.

The spectral properties of Schrödinger operators for periodic systems (crystals) are well-understood, thanks to Bloch-Floquet theory. When such periodic systems are cut in half, an "edge spectrum" can appear. This spectrum describes physical properties appearing at the boundary of the system. In this talk, we present a general framework to study this edge spectrum. We make the connection with topological insulators (and bulk-edge correspondence), and we also prove that, for any material cut with an irrational angle, its edge spectrum fills all the bulk gaps.

We study the two-dimensional two-component plasma or Coulomb gas. We obtain free energy expansions and Large Deviations Principles which reveal the well-known KT transition from a system with free charges to a system with bound dipoles. Based on joint works with Jeanne Boursier, Thomas Leblé, Ofer Zeitouni.

Many physical situations are modelled by nonlinear equations, from Einstein's equations to nonlinear elasticity and nonlinear acoustics. Inverse problems associated with these equations consist in determining the coefficients of the equations from measured data. We show that the nonlinearity helps in solving these inverse problems in cases that we don't know how to solve the corresponding linearized problem.

In a typical setting, the energy levels of an electron moving in a magnetic field are eigenvalues of a special magnetic Laplace operator involving the semiclassical parameter (a very small parameter compared to the sample’s scale). Searching for asymptotic expansions of these eigenvalues, in the singular limit of vanishing semiclassical parameter, is a rich topic that will be partially reviewed, in the present talk. I’ll discuss recent results involving magnetic fields varying on very short scales (modeled by piecewise constant functions with discontinuity along curves), for which some asymptotics are derived via a purely variational proof not involving pseudo-differential calculus.

We consider a quantum spin system in any dimension which has a spectral gap above its ground state energy. We assume that the ground state is frustration-free, locally disordered and does not exhibit long range entanglement. We prove that arbitrary perturbations at the boundary of a large domain do not affect ground state expectation values inside the domain. The proof relies on an isoperimetric inequality for the ground state energy. This is joint work with Wojciech De Roeck, Brecht Donvil, and Martin Fraas.

The problem of determining the local eigenvalue statistics (LES) for one-dimensional random band matrices (RBM) will be discussed with an emphasis on the localization regime. RBM are real symmetric $(2N+1) \times (2N+1)$ matrices with nonzero entries in a band of width $2 N^\alpha +1$ about the diagonal, for $0 \leq \alpha \leq 1.$ The nonzero entries are independent, identically distributed random variables. It is conjectured that as $N \rightarrow \infty$ and for $0 \leq \alpha '< \frac{1}{2}$, the LES is a Poisson point process, whereas for $\frac{1}{2} '< \alpha \leq 1$, the LES is the same as that for the Gaussian Orthogonal Ensemble. This corresponds to a phase transition from a localized to a delocalized state as $\alpha$ passes through $\frac{1}{2}$. In recent works with B. Brodie and with M. Krishna, we have made progress in proving this conjecture for $0 \leq \alpha '< \frac{1}{2}$.
Some of the results by others for the delocalized state with $\frac{1}{2} '< \alpha \leq 1$ will also be described.

Estimating the location and accumulation rate of eigenvalues of Schrödinger operators is a classical problem in spectral theory and mathematical physics. The pioneering work of R. Frank (Bull. Lond. Math. Soc., 2011) illustrated the power of Fourier analytic methods - like the uniform Sobolev inequality by Kenig, Ruiz, and Sogge, or the Stein-Tomas restriction theorem - in this quest, when the potential is non-real and has "short range".

Recently S. Bögli and J.-C. Cuenin (arXiv:2109.06135) showed that Frank's "short-range" condition is in fact optimal, thereby disproving a conjecture by A. Laptev and O. Safronov (Comm. Math. Phys., 2009) concerning Keller-Lieb-Thirring-type estimates for eigenvalues of Schrödinger operators with complex potentials.

In this talk, we estimate complex eigenvalues of continuum random Schrödinger operators of Anderson type. Our analysis relies on methods of J. Bourgain (Discrete Contin. Dyn. Syst., 2002, Lecture Notes in Math., 2003) related to almost sure scattering for random lattice Schrödinger operators, and allows us to consider potentials which decay almost twice as slowly as in the deterministic case.

The talk is based on joint work with Jean-Claude Cuenin.

We study the time dependent Schrödinger equation for large spinless fermions with the semiclassical scale $h = N^{-1/3}$ in three dimensions. By using the Husimi measure defined by coherent states, we rewrite the Schrödinger equation, with regularized Coulomb with a polynomial cutoff depending on $N$, into a Vlasov type equation with semiclassical and mean field remainder terms. In case of repulsive Coulomb interaction, by using the uniform estimates of kinetic energy and the localized number operator, the convergence of both remainder terms has been obtained, and the convergence to Vlasov Poisson equation is proceeded with additional compactness argument. The talk is based on the joint works with M. Liew and J. Lee.

We will consider the time evolution of initially confined fermions interacting through a singular potential. In a joint mean-field and semiclassical scaling, we will show that the many-body dynamics can be approximated in Schatten norms by the time-dependent Hartree-Fock equation. Our result holds for mixed states enjoying a semiclassical structure. Moreover, in the semiclassical limit, the dynamics is close to the one described by the Vlasov equation with singular interactions, thus providing a two-step approximation of the many-body evolution. The method used provides explicit bounds on the convergence rate.

Based on joint works with J. Chong and L. Laflèche.

We consider a Dirac system confined to a domain $\Omega$ in the plane and subject to a perpendicular magnetic field. In this talk I will present results on accurate asymptotic estimates for the low-lying eigen-energies in the limit of strong magnetic field under fairly general local boundary conditions. We will focus on the case of infinite mass boundary conditions. In particular, we will see the emergence of exceptional energies, very stable in the geometry, which can be heuristically understood by looking at the half-plane problem.

This talk is based on joint work with Jean-Marie Barbaroux (Université de Toulon), Loic Le Treust (Aix Marseille Univ, France) and Nicolas Raymond (Université d’Angers); arXiv: 1810.03344 (JEMS '21) and arXiv: 2007.03242.

We establish a precise relation between the integral kernel of the scattering matrix and the resonance in the Spin-Boson model, which describes the interaction of a two-level quantum system with a second-quantized scalar field. For this purpose, we derive an explicit formula for the two-body scattering matrix. This provides a mathematical framework for the peaks at certain energy values in the scattering cross-section that physicists have observed since the beginning of quantum mechanics. In the case of quantum mechanics, this problem was addressed in numerous works that culminated in Simon’s seminal paper (Ann Math, 1973). However, in quantum field theory, the analogous problem was open for several decades at the level of rigor required in mathematics.

We study the ground-state of a gas of N Bosons in ${\mathbb R}^3$ interacting via three-body interactions in the Gross-Pitaevskii regime. In this regime the interaction potential scales as $N V(N^{1/2}(x-y,x-z))$ and is so singular that the mean-field approximation fails to describe the system at main order. We will see that the three-body correlation structure creates a shift in energy of order $N$ and we will explain how to describe it. The effective theory is then given by the 3D energy-critical non-linear Schrödinger functional. This is joint work with P.T. Nam and J. Ricaud.

We consider 2D quantum materials, modeled by a continuum Schrödinger operator whose potential is an array of identical potential wells centered on the vertices of discrete subset, $\Omega$, of the plane (not necessarily translation invariant). We discuss the regime of deep wells (strong binding), where the spectrum is approximated by that of a discrete (tight-binding) Hamiltonian. In the translation invariant case, our results imply the convergence of low-lying Floquet-Bloch dispersion surfaces to those of the limiting tight-binding operator. Examples include the single electron model for bulk graphene ($\Omega$=honeycomb lattice), and a sharply terminated graphene half-space, interfaced with the vacuum along an arbitrary line-cut.

We also apply our approach to the study the case of a strong constant perpendicular magnetic field. In this case, strong resolvent convergence is used to prove the equality of topological indices (for bulk and edge systems) of discrete (tight binding) and continuum (strongly bound) Hamiltonians.

Finally, we present recent results on the spectrum of the tight-binding Hamiltonian obtained from the sharp termination of a graphene half-space, interfaced with the vacuum along an arbitrary rational line-cut.

This lecture focuses on joint works with C.L. Fefferman, S. Fliss and J. Shapiro.

Motivated by the study of high energy Steklov eigenfunctions, we examine the semi-classical Robin Laplacian. In the two dimensional situation, we determine an effective operator describing the asymptotic distribution of the negative eigenvalues, and we prove that the corresponding eigenfunctions decay away from the boundary, for all dimensions.

We provide an upper bound on the ground state energy per unit volume of a dilute Bose gas in the thermodynamic limit capturing the correct second order term, as predicted by the celebrated Lee-Huang-Yang formula. With respect to the first proof of this result, given by H.-T. Yau and J. Yin, our trial state is simpler, it applies to a larger class of interactions and it provides a better rate.

Joint work with S. Cenatiempo and B. Schlein.

The Kac master equation provides a simple framework for understanding equilibration of particle systems. In this talk I'll review results on the entropy and information decay for a one dimensional Kac system coupled to a reservoir as well as results on the gap of the generator of a Kac process describing a three dimensional gas of colliding particles. This is joint work with Federico Bonetto, Eric Carlen and Maria Carvalho

In this talk we discuss quantitative improvements for Weyl remainders under dynamical assumptions on the geodesic flow. We consider a variety of Weyl type remainders including asymptotics for the eigenvalue counting function as well as for the on and off diagonal spectral projector. These improvements are obtained by combining the geodesic beam approach to understanding eigenfunction concentration together with an appropriate decomposition of the spectral projector into quasimodes for the Laplacian. One striking consequence of these estimates is a quantitatively improved Weyl remainder on all product manifolds. This is joint work with Y. Canzani.

After giving an introduction to Szegö-type asymptotics for Schrödinger operators, we discuss recent progress and present a new result on the stability of such an asymptotic expansion under perturbations: We show that, for a fairly large class of test functions, the second-order Szegö-type asymptotics for the multi-dimensional Laplacian, which was proved by Leschke, Sobolev and Spitzer in 2014, does not change upon perturbing the Laplacian by a compactly supported bounded potential. In our proof, we control the effect of the perturbation with the help of a limiting-absorption principle. The result has also implications for entanglement entropies of ground states of quasi-free Fermi gases.

We consider a model for two types (bath and tracers) of 2D quantum particles in a perpendicular (artificial) magnetic field. Interactions are short range and only inter-species, and we assume that the bath particles are fermions, all lying in the lowest Landau level of the magnetic field. Heuristic arguments then indicate that, if the tracers are strongly coupled to the bath, they effectively change their quantum statistics, from bosonic to fermionic or vice-versa. We rigorously compute the energy of a natural trial state, indeed exhibiting this phenomenon of statistics transmutation. The proof is based on (seemingly ?) new estimates for the characteristic polynomial of the Ginibre ensemble of random matrices.

Work in progress with Gaultier Lambert and Douglas Lundholm.

In this talk we consider a magnetic Schrödinger operator H in a domain $\Omega \subset {\mathbb R}^2$ with compact boundary. We impose Dirichlet boundary conditions on $\partial \Omega$. For a constant magnetic field having large intensity, we focus on the existence and the description of the edge states , namely eigenfunctions for H whose mass is localized along the boundary $\partial \Omega$. We show that such edge states exist and we give a detailed description of the localization and distribution of their mass along $\partial \Omega$. From this result, we also infer asymptotic formulas for the eigenvalues of H. If time allows, we briefly discuss how the previous localization results generalise to a class of Iwatsuka models , namely when the presence of a boundary $\partial \Omega$ is replaced by a fast oscillation of the magnetic field along an interface.

This talk is based on joint works with J.J. L. Velazquez (IAM Bonn).

This talk concerns the nonlinear (massive) Dirac equation with a nonlinearity taking the form of a space-dependent mass, known as the (generalized) Soler model. The equation has standing wave solutions for frequencies w in (0,m), where m is the mass in the Dirac operator. These standing waves are generally expected to be stable (i.e., small perturbations in the initial conditions stay small) based on numerical simulations, but there are very few results in this direction. The results that I will discuss concern 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 present some partial results for the one-dimensional case, highlight the differences and similarities with the Schrödinger case, and discuss (a lot of) open problems.

This is joint work with Danko Aldunate, Julien Ricaud and Edgardo Stockmeyer.

We will discuss Weyl laws with a sharp remainder term, in the 'integrated' (counting eigenvalues) and 'pointwise' (concerning the ground state density) forms. We will explain how the presence of a singular potential modifies the standard asymptotics. Since the microlocal approach of Levitan and Hörmander is not obvious to adapt in this setting, we will show how the older proof due to Avakumovic is well suited to include singular potentials. This is a joint work with Rupert Frank.

We study the low-temperature asymptotics of the free energy of the quantum Heisenberg ferromagnet, and show that it agrees to leading order with the one of an ideal Bose gas of magnons at criticality, as predicted by the spin-wave approximation. For the model in 3 (or more) dimensions, this was shown a few years ago (in joint work with Correggi and Giuliani), but the analysis is significantly more complicated in lower dimensions. In fact, the two-dimensional case is still open, but the one-dimensional case was recently settled (in joint work with Napiorkowski). We will explain the main ingredients in the proof.

The empirical spectral distribution of a non-selfadjoint random matrix concentrates around a deterministic probability measure on the complex plane as its dimension increases. Despite the inherent spectral instability of such models, this approximation is valid all the way down to local scales just above the typical eigenvalue spacing distance. We will present recent results on eigenvalue spectra for non-selfadjoint random matrices with correlated entries and their application to systems of randomly coupled differential equations that are used to model a wide range of disordered dynamical systems ranging from neural networks to food webs.

The theory of spin transport in quantum systems, as compared to charge transport, is still in a preliminary stage. Whenever the spin operator does not commute with the unperturbed Hamiltonian operator, as it happens in the so-called Rashba insulators , the lack of commutativity poses technical and conceptual problems for the theory.

In my talk, I will first address some foundational questions in spin transport theory. Then, I will present a new formula for the transverse spin conductivity in gapped (periodic) quantum systems, which generalizes to spin the celebrated double-commutator formula for charge conductivity, sometimes dubbed Kubo-Chern formula. As a corollary, one obtains that whenever the spin is (almost) conserved, the transverse spin conductivity is (approximately) equal to the spin-Chern number.

The method of proof relies on the characterization of a non-equilibrium almost-stationary state (NEASS), which well approximates the physical state of the system (in the sense of space-adiabatic perturbation theory) and allows moreover to compute the response of the adiabatic spin current as the trace per unit volume of the spin current operator times the NEASS.

The talk is based on results obtained jointly with G. Marcelli and S. Teufel, and with G. Marcelli and C. Tauber.

The Ginzburg-Landau model is a phenomenological description of superconductivity. A crucial feature is the occurrence of vortices (similar to those in fluid mechanics, but quantized), which appear above a certain value of the strength of the applied magnetic field called the first critical field. In this talk I will present a sharp estimate of this value and describe the behavior of global minimizers for the 3D Ginzburg-Landau functional below and near it. This is partially joint work with Etienne Sandier and Sylvia Serfaty.

The famous van der Waals law shows that neutral atoms attract each other over large distances with an effective potential of the form $-c/d^6$ , $d$ the distance of the atoms. There are additional higher order terms, which depend on binary atom-atom interactions at large distances. The Axilrod-Teller-Muto correction, which is the first term which depends on triplets of atoms and is of importance in atom physics., shows up at order $d^9$. We prove that the Axilrod-Teller-Muto conjecture is correct. This is joint work with Jean-Marie Barbaroux, Michael Hartig, and Semjon Wugalter

We showed that the (Gaertner-Ellis) logarithmic moment generating function for Born measures associated with KMS states of weakly interacting fermions on the lattice can be written as the limit of logarithms of Gaussian Berezin integrals. This representation can be used to prove analyticity (near the origin) of the generating function, yielding, in particular, a large deviation principle for the Born measures. The construction does not use translation invariance and also random background potentials can be considered. A possible and important extension of this result, in view of applications to the macroscopic behavior of the electric conductivity at nanoscales, in presence of interactions, will be discussed. Recent results on the macroscopic behavior at nanoscales for free fermions with disorder will be presented to motivate the study of large deviation principles for the Born measures of lattice fermions.

Homogeneous Schrödinger operators, called also Bessel operators are given by $H_m=-\partial_x^2+(-\frac14+m^2)\frac{1}{x^{2}},$ with the boundary condition $\sim x^{m+\frac12}$ near $0$. I will discuss their role in representations of the smallest semi-simple non-compact Lie group $\mathrm{SL}(2,\mathbb{R})$.

I will discuss a novel type of a derivative nonlinear Schrödinger type equation, which can be seen as a continuum limit of completely integrable Calogero-Moser systems. In this talk, I will focus on dynamics of $N$-soliton solutions, turbulence (growth of Sobolev norms), and a Lax pair structure. This is ongoing joint work with Patrick Gérard (Orsay).

We study the time evolution of the strongly coupled polaron described by the Fröhlich Hamiltonian with large coupling constant $\alpha$. For initial data of Pekar product form with coherent phonon field and with sufficiently small energy, we provide a norm approximation of the time evolution valid for all times of order $\alpha^2$. The approximation is given in terms of a Pekar product state, evolved through the Landau-Pekar equations, corrected by a Bogoliubov dynamics taking quantum fluctuations into account. The proof is based on an adiabatic theorem for the Landau-Pekar equations and the persistence of the spectral gap.

This is joint work with D. Feliciangeli, N. Leopold, D. Mitrouskas, B. Schlein and R. Seiringer.

The one-particle density matrix $\gamma(x; y)$ is one of the key objects in the quantum-mechanical approximation schemes. The self-adjoint operator $\Gamma$ with the kernel $\gamma(x; y)$ is trace class but a sharp estimate on the decay of its eigenvalues was unknown. In this talk I will present a sharp bound and an asymptotic formula for the eigenvalues of $\Gamma$.

We will present an overview of extremal eigenvalues of the Robin Laplacian. This includes recent results for low eigenvalues and, time permitting, the high-frequency behaviour and its effect on a Pólya-type conjecture. Based on joint work with Pedro Antunes, James Kennedy, David Krejčiřík, and Richard Laugesen.

NOTICE: This talk is an hour later than usual (at 5 p.m. CET due to the time difference with California).

Lieb-Thirring inequalities bound sums of powers of eigenvalues of Schrödinger operators in terms of integrals of the potential. We survey the problem of sharp constants in these inequalities and discuss recent progress. We include some negative results on the so-called one-particle constant, obtained jointly with Gontier and Lewin.

We consider systems of N bosons trapped in a two-dimensional box with volume one, interacting through a repulsive potential with scattering length exponentially small in N. We show that in this scaling, known as the two dimensional Gross-Pitaevskii regime, low-energy states exhibit Bose-Einstein condensation with an explicit (optimal, up to logarithmic corrections) bound on the number of orthogonal excitations. The main challenge to be overcome is that in the two dimensional Gross-Pitaevskii regime correlations among the particles are so strong that the ratio between the integral of the potential and the effective coupling appearing in the expressions for the thermodynamic functions is of order N. Joint work with C. Caraci and B. Schlein.

It is well known that Schrödinger operator with degenerate kinetic energies may support infinitely many weakly coupled bound states. Hainzl and Seiringer proved that the eigenvalues of $|p^2-1|-\lambda V$ are exponentially small as the coupling constant $\lambda$ tends to zero, and they connected the problem to an effective operator on the sphere. This holds essentially under the assumption that the potential is integrable. In this talk I will show that one can relax this condition considerably and cover a much larger class of potentials. This is joint work with Konstantin Merz.

The talk will explain how the bulk-boundary correspondence can be approached from a purely PDE point of view, inspired by a recent work of A. Drouot ( https://arxiv.org/abs/1909.10474 ). We will derive a key formula which is valid at non-zero temperature, in the presence of long-range magnetic fields, and without any gap conditions. The usual bulk-boundary correspondence is then obtained by taking the zero temperature limit and assuming a gap condition. An important rôle in the proof of this limit is played by the so-called Streda formula https://arxiv.org/abs/1810.05623

This is joint ongoing work with M. Moscolari (Aalborg) and S. Teufel (Tübingen).

In this talk I will present various results concerning interpolation inequalities, best constants and information about the extremal functions concerning Schrödinger magnetic operators in dimensions 2 and 3. The particular, and physical interesting cases of constant and of Aharonov-Bohm magnetic fields will be discussed in detail.

These works have been made in collaboration with D. Bonheure, J. Dolbeault, A. Laptev and M. Loss.

This talk is devoted to the spectral analysis, in the strong magnetic field limit, of the 2D Dirac operator on a smooth open bounded domain of the plane whose boundary carries the 'MIT bag' condition.

We establish the one-term asymptotic expansions of the first positive eigenvalues, and of the first negative eigenvalue.

This is a joint work with J-M. Barbaroux, L. Le Treust, and E. Stockmeyer.

I will discuss the Bogolubov variational model that arises when a many-body bosonic Hamiltionian is restricted to quasi-free states. For simplicity I will restrict attention to ground states. By definition the variational model gives an upper bound on the many-body ground state energy. I will discuss examples where it is, in fact, a good approximation, such as the Lieb-Liniger model, the one- and two-component charged Bose gases and the dilute Bose gas in three dimensions.

The Gagliardo-Nirenberg-Sobolev (GNS) inequalities have played a major role in applied mathematics and mathematical physics in more than half a century. In this talk I will present several new results concerning the counterparts of the GNS inequalities on bounded domains. Of course concentration of the minimizers of the GNS inequalities is a main tool in the proof of existence of minimizers on bounded domains. Naturally concentration occurs in the interior for Dirichlet boundary conditions and on the boundary for Neumann boundary conditions. In the Neumann case this leaves a set of interesting open problems depending on the characteristic of the boundary.

This is in part joint work with Cristobal Vallejos (PUC) and Hanne Van Den Bosch (U. de Chile) and in part with Soledad Benguria (U. Wisconsin, Madison).