Workshop II
May 23 - 27
2022
Schedule
Schedule
Speakers
Speakers
Politecnico di Torino
Ground states for the NLS with a Dirac’s delta in dimension two and three
[video]Widely studied in dimension one and in branched structures, the dynamics of the Nonlinear Schroedinger Equation in the presence of a point interaction becomes in higher dimension more complicated even in the rigorous formulation of the problem. Here we discuss the existence and the symmetry features of the Ground States, namely the minimizers of the energy at fixed mass.This is a joint work with Filippo Boni, Raffaele Carlone, and Lorenzo Tentarelli.
Università dell'Insubria
Semiclassical evolution of Gaussian states in presence of singular interactions
[video] [slides]I will present several results aimed at describing the semiclassical approximation of the quantum evolution of one-dimensional Gaussian coherent states in presence of singular potentials. I will show semiclassical approximation both of the quantum evolution at finite time and of the scattering operators. The generator of the semiclassical dynamics approximating the quantum dynamics is obtained as a singular perturbation of the free Liouville operator. This work is a joint collaboration with Davide Fermi and Andrea Posilicano.
Politecnico di Milano
Derivation of the Ginzburg-Landau Theory for Interacting Fermions in a Trap
[video] [slides]We study a dilute gas of fermions, at temperature $T=0$ and chemical potential $\mu \in \mathbb{R}$. The particles are trapped by an external harmonic potential, and they interact via a microscopic attractive two-body potential. We prove the emergence of the macroscopic Ginzburg-Landau theory as first-order contribution to the BCS energy functional in the regime of vanishing micro-to-macro scale parameter. Joint work with M. Correggi.
University of Zurich
Microscopic Derivation of Ginzburg–Landau Theory and the BCS Critical Temperature Shift in a Weak Homogeneous Magnetic Field
[video] [slides]Starting from the Bardeen–Cooper–Schrieffer (BCS) free energy functional, we derive the Ginzburg–Landau functional for the case of a weak homogeneous magnetic field. We also provide an asymptotic formula for the BCS critical temperature as a function of the magnetic field. This extends the previous works of Frank, Hainzl, Seiringer and Solovej to the case of external magnetic fields with non-vanishing magnetic flux through the unit cell. Joint work with Marcel Schaub and Christian Hainzl.
Université Paris Est
Semi-classical analysis in a subelliptic context
[video] [slides]We will present recent results obtained in collaboration with Véronique Fischer and Steven Flynn, both from the University of Bath. We will explain how to use the harmonic analysis of non-commutative nilpotent graded Lie groups to develop a semi-classical approach adapted to problems involving subelliptic operators. We will describe applications to the properties of sequences of eigenfunctions of a sub-Laplacian on a nil-manifold.
Aarhus Universitet
The ground state energy of the dilute Bose gas
[video] [slides]In this talk, I will review recent work on the energy of a dilute Bose gas in the thermodynamic limit in 2 and 3 dimensions. A key example is the case of hard core interactions, for which the lower bound including the first correction term - the Lee-Huang-Yang term - was recently established in the 3 dimensional case. It turns out that problem faced in the 2 dimensional case, even for regular potentials, is very analogous to the hard core case in 3 dimensions. The talk will illustrate the similarities and differences between these settings.Based on joint work with J.P.Solovej as well as T. Girardot, L. Junge, L. Morin and M. Olivieri.
LMU Munich
A 3D-Schrödinger operator under magnetic steps
[video] [slides]We consider Schrödinger operators on the space and the half-space with discontinuous magnetic fields having a piecewise-constant strength and a uniform direction. Motivated by applications in the theory of superconductivity, we study the infimum of the spectrum of these operators. Working in the half-space, we further give sufficient conditions on the strength and the direction of the magnetic field such that the aforementioned infimum is an eigenvalue of a reduced model operator on the half-plane. Joint work with W. Assaad.
Michal Jex
Michal Jex
CEREMADE - Université Paris Dauphine
Quantum Systems at the Brink: Helium-Type Systems
[video] [slides]We study two challenging problems for helium–type systems. Existence of eigenvalues at thresholds and the asymptotic behaviour of the corresponding eigenfunctions. Since the usual methods for addressing these problems need a safety distance to the essential spectrum, they cannot be applied in critical cases, when an eigenvalue enters the continuum. We develop a method to address both problems and derive sharp upper and lower bounds for the asymptotic behaviour of the ground state of critical helium–type systems at the threshold of the essential spectrum. This is the first proof of the precise asymptotic behaviour of the ground state for this benchmark problem in quantum chemistry. Moreover, our bounds describe precisely how the asymptotic decay of the ground state changes, when the system becomes critical. In addition, we show the existence of a ground state of this quantum critical system with a finite nuclear mass. Previously this had been known only in the Born–Oppenheimer approximation of infinite nuclear mass.
Lebanese University
On the Isoperimetric Inequality for the Magnetic Robin Laplacian with Negative Boundary Parameter
[video] [slides]Among domains with the same area, the disk is a minimiser of the magnetic Dirichlet ground-state energy, provided that the magnetic field is homogeneous. The question seems challenging when imposing a Neumann boundary condition, where spectral asymptotics suggest that the disc is a potential maximiser. Imposing a Robin condition, the sign of the boundary parameter will play a role; when it is negative, the disk maximises the ground state energy among a wide class of domains with a given perimeter, provided that the magnetic field is of moderate strength. In the other cases, the question will be illustrated by discussing the known spectral asymptotics. The talk is based on a joint work with Vladimir Lotoreichik.
SISSA, Trieste
Quantum Systems at The Brink: Existence of Bound States, Critical Potentials and Dimensionality
[video] [slides]The existence of bound states plays a crucial role for the properties of quantum systems. We present a necessary and sufficient condition for Schrödinger operators to have a zero energy bound state. In particular we show that the asymptotic behavior of the potential is the crucial ingredient. The existence and non-existence result complement each other and exhibit a strong dependence on the dimension.
BCAM, Bilbao
Implementing Bogoliubov transformations beyond the Shale-Stinespring condition
[video] [slides]Quantum many-body systems can be mathematically described by vectors in a certain Hilbert space, the so-called Fock space, whose Schrödinger dynamics are generated by a self-adjoint Hamiltonian operator H. Bogoliubov transformations are a convenient way to manipulate H while keeping the physical predictions invariant. They have found widespread use for analyzing the dynamics of quantum many-body systems and justifying simplified models that have been heuristically derived by physicists. In the 1960's, Shale and Stinespring derived a necessary and sufficient condition for when a Bogoliubov transformation is implementable on Fock space, i.e., for when there exists a unitary operator U such that the manipulated Hamiltonian takes the form U* H U. However, non-implementable Bogoliubov transformations appear frequently in the literature for systems of infinite size. In this talk, we therefore construct two extensions of the Fock space on which certain Bogoliubov transformations become implementable, although they violate the Shale-Stinespring condition.
University of Bari & INFN
Generalized spin-boson models with non-normalizable form factors
[slides]Generalized spin-boson (GSB) models describe the interaction between a quantum mechanical system and a structured boson environment, mediated by a family of coupling functions known as form factors. We propose an extension of the class of GSB models which can accommodate non-normalizable form factors, provided that they satisfy a weaker growth constraint, thus accounting for a rigorous description of a wider range of physical scenarios; we also show that such "singular" GSB models can be rigorously approximated by GSB models with normalizable form factors. Furthermore, we discuss in greater detail the structure of the spin-boson model with a rotating wave approximation (RWA): for this model, the result is improved via a nonperturbative approach which enables us to further extend the class of admissible form factors, as well as to compute its resolvent and characterize its self-adjointness domain.
Università di Milano
Anomalies and non-perturbative QFT
[video] [slides]Adler and Bardeen in 1969 established the non-renormalization of the chiral anomaly, writing it as a perturbative expansion which is order by order vanishing; since then, this property has found uncountable applications, including the Standard Model consistence via the anomaly cancellation condition (Bouchiat, Iliopoulos, Meyer 1972). After reviewing briefly this notion, we prove the anomaly non renormalization at a non-perturbative level in the case of lattice vector boson-fermion models in d=1+1 (uniformly in the lattice) and in d=3+1 (up to a cut-off of the order of the inverse coupling). The proof relies on bounds on the large distance decay of correlations and Ward Identities, both exact and emerging. In the case of chiral models, like the effective electroweak theory with quartic Fermi interaction in d=3+1, some Ward Identites are violated and the anomaly can be proved to vanish up to subdominant corrections which are rigorously bounded, under the cancellation condition on charges and with a cut-off of the order of the inverse coupling. Open problems and conjectures will be finally discussed.
David Mitrouskas
David Mitrouskas
IST Austria
Ground state energy of the weakly coupled 2D Fermi polaron
[video] [slides]The Fermi polaron is a popular physics model for the description of an impurity interacting with a gas of fermions via two-body zero-range interaction. We show that in the limit of weak coupling, the ground state energy of this system is described by the so-called polaron energy. The polaron energy is the optimal energy estimate obtained from trial states up to first order in particle-hole expansion, which was proposed by F. Chevy in the physics literature.
Aarhus Universitet
Eigenvalue asymptotics for confining magnetic Schrödinger operators with complex potentials
[video]This talk is devoted to the spectral analysis of the electro-magnetic Schrödinger operator with complex potential. In the semiclassical limit, we derive a pseudo-differential effective operator that allows us to describe the spectrum in various situations and appropriate regions of the complex plane. Not only results of the selfadjoint case are proved (or recovered) in the proposed unifying framework, but new results are established when the electric potential is complex-valued. In such situations, when the non-selfadjointness comes with its specific issues (lack of « spectral theorem », resolvent estimates), the analogue of the « low-lying eigenvalues » of the selfadjoint case are still accurately described and the spectral gaps estimated.
IST Austria
Energy-momentum relation of the strongly coupled polaron
[video] [slides]The polaron model is a basic model of quantum field theory describing a single particle interacting with a bosonic field, where it is expected that the field behaves purely classically at strong coupling. We provide evidence for this in the context of the ground state energy of the system at fixed non-zero momentum by proving appropriate lower and upper bounds for the UV-regularized model at strong coupling. For the polaron model in a polar crystal which exhibits an apparent UV divergence, we provide the corresponding upper bound. In this case, also the quantum fluctuations of the field need to be taken into account.
LAGA - Université Paris Nord
Boundary conditions for Bismut's hypoelliptic Laplacian
[video] [slides]In this joint work with Shu Shen we established that natural boundary conditions, which extend to the case of Bismut's hypoelliptic Laplacian, the Dirichlet and Neumann boundary conditions for Hodge Laplacian, allow the commutation of closed realization of the deformed differential (and codifferential) with the resolvent of the hypoelliptic Laplacian.
Università di Milano Bicocca
The nonlinear Schrödinger equation with isolated singularities
[video] [slides]The nonlinear Schrödinger (NLS) equation with a point interaction and power nonlinearity in dimension two and three is introduced and its well posedness is considered. Behind the autonomous interest of the problem, this is a model of the evolution of so-called singular solutions that are well known for a long time in the analysis of semilinear elliptic equations. We show that the Cauchy problem enjoys local existence and uniqueness of strong solutions, and that the solutions depend continuously from initial data. In dimension two, well posedness holds for any power nonlinearity and global existence is proved for powers below the cubic. In dimension three local and global well posedness are restricted to low powers. Some further open problems and directions will be discussed.Work in collaboration with Claudio Cacciapuoti and Domenico Finco.
ENS Lyon
Thomas-Fermi profile of a fast rotating Bose-Einstein condensate
[video] [slides]We study the minimizers of a magnetic 2D non-linear Schrödinger energy functional in a quadratic trapping potential, describing a rotating Bose-Einstein condensate. We derive an effective Thomas-Fermi-like model in the rapidly rotating limit where the centrifugal force compensates the confinement, and available states are restricted to the lowest Landau level. The coupling constant of the effective Thomas-Fermi functional is to linked the emergence of vortex lattices (the Abrikosov problem). We define it via a low density expansion of the energy of the corresponding homogeneous gas in the thermodynamic limit.
Aarhus Universitet
Casimir-Polder effect for an atom interacting with a conductor wall
[video] [slides]We study a system composed of a hydrogen atom interacting with an infinite conductor wall. The interaction energy decays like L^{-3}, where L is the distance between the atom and the wall, due to the emergence of the Van der Waals forces. In this talk we show how, considering the contributions from the quantum fluctuations of the electromagnetic field, the interaction is weakened to a decay of order L^{-4} giving rise to the retardation effects which fall under the name of Casimir-Polder effect.
Università dell'Insubria
On the origin of Minnaert resonances
[video] [slides]We consider the appearance of what are called "MInnaert resonances" in the scattering of sound waves in a medium with a small inhomogeneity enjoying a high contrast of both its mass density and bulk modulus. This phenomenon is explained in terms of the behaviour, as the size of the inhomogeneity decreases to zero, of the norm resolvent limit of a related frequency-dependent Schroedinger operator, the limit being not trivial if and only if the frequency coincides with that of Minnaert.This is a joint work with Andrea Mantile and Mourad Sini.
Université d'Angers
Semiclassical analysis of the Neumann Laplacian with constant magnetic field in three dimensions
[video]This talk deals with the spectral analysis of the semiclassical Neumann magnetic Laplacian on a smooth bounded domain in dimension three. When the magnetic field is constant and in the semiclassical limit, we establish a four-term asymptotic expansion of the low-lying eigenvalues, involving a geometric quantity along the apparent contour of the domain in the direction of the field. In particular, we prove that they are simple under generic assumptions and we are led to revisit the two-term expansion of the lowest eigenvalue obtained by Helffer and Morame in 2004. Joint work with Frédéric Hérau.
Università di Milano Bicocca
Quantum Vortex Dynamics by Geometric and Topological Methods
[video] [slides]Università di Modena e Reggio Emilia
Nonlinear Stark-Wannier equations
[video] [slides]In this talk we discuss some recent results for a class of nonlinear models in Quantum Mechanics. In particular we focus our attention to the nonlinear one-dimensional Schrödinger equation with a periodic potential and a Stark-type perturbation. Concerning stationary states we prove that, in the limit of large periodic potential, the Stark-Wannier ladders of the linear equation become a dense energy spectrum because a cascade of bifurcations of stationary solutions occurs when the ratio between the effective nonlinearity strength and the tilt of the external field increases. Furthermore we show that in the same limit the time behavior of the wavefunction can be approximated, with a precise estimate of the remainder term, by means of the solution to the discrete nonlinear Schrödinger equation of the tight-binding model.
Politecnico di Torino
Symmetry Breaking in Two-dimensional Square Grids
[video] [slides]In the talk, I will present some recent results on the model robustness of the infinite two-dimensional square grid with respect to symmetry breakings due to the lack of finitely or infinitely many edges. Precisely, I will discuss how these topological perturbations of the square grid affect the so-called dimensional crossover identified in [Adami et al., ADPE, 2019]. Such a phenomenon has two evidences: the coexistence of the one and the two-dimensional Sobolev inequalities and the appearence of a continuum of L^2-critical exponents for the ground states at fixed mass of the focusing nonlinear Schrödinger equation. I will show which classes of defects do preserve the dimensional crossover and which classes do not, from this twofold perspective.