Numerical and theoretical advances in quantum mechanics

Liste des contributions

3. Mario Pulvirenti: "Some considerations on the derivation of the Quantum Boltzmann equation"

16/01/2023 10:30

In this talk I consider the problem of the derivation of the nonlinear Boltzmann equation from a quantum particle system in the weak-coupling limit, the very few rigorous results, the difficult open problems and possible perspectives.

6. Claudia Negulescu: "Decoherence rhapsody in the photosynthesis process"

16/01/2023 11:30

It is said that classical theories are sometimes inappropriate to describe very efficient biological processes in nature, which seem to be better understood via quantum mechanical models. We are however still very far from understanding how quantum features can survive in open quantum systems. In this talk I shall present two simple mathematical models for the illustration of the excitation...

2. Clotilde Fermanian-Kammerer: Semiclassical gaussian states and dynamical approximation of quantum systems

clotilde fermanian

16/01/2023 14:30

In this talk we discuss methods for approximating of the solution of a semi-classical Schrödinger equation. Originated in a theoretical chemistry context, these methods have in common to strongly on the use of semiclassical Gaussian states.

4. Peter Müller: "Entanglement entropy for quasifree Fermi gases: area law versus logarithmic enhancement"

16/01/2023 15:30

An overview is presented of Szeg\H{o}-type asymptotics for spectral projections of multi-dimensional continuum Schr\"odinger operators. Whenever possible we treat general test functions, including those which describe entanglement entropies of corresponding quasifree Fermi gases. Unfortunately, there exists no general theory which allows to deduce the leading asymptotic behaviour of...

5. David Krejcirik: Is the optimal rectangle a square?

16/01/2023 17:00

We give a light talk on an optimality of a square in geometry and physics. First, we recollect classical geometric results that the square has the largest area (respectively, the smallest perimeter) among all rectangles of a given perimeter (respectively, area). Second, we recall that the square drum has the lowest fundamental tone among all rectangular drums of a given area or perimeter and...

7. Anton Arnold: "A hybrid WKB-based method for Schrödinger scattering problems in the semi-classical limit"

17/01/2023 09:00

We consider 1D scattering problems related to quantum transport in diodes. We discuss the efficient numerical integration of ODEs like epsilon^2u"+a(x)u=0 for 0<epsilon<<1 on coarse grids, but still yielding accurate solutions; including oscillatory (for given a(x)>0) and evanescent regimes (for a(x)<0), partly including turning points. In the oscillatory case we use a marching method...

8. Paola Pietra: "An interface formulation for the Poisson equation in the presence of a semiconducting single-layer material"

17/01/2023 10:30

We consider a semiconducting device with an active zone made of a single-layer material. The associated Poisson equation for the electrostatic potential (to be solved in order to perform self-consistent computations) is characterized by a surface particle density and an out-of-plane dielectric permittivity in the region surrounding the single-layer. To avoid mesh refinements in such a region,...

9. Christian Lubich: "Time integration of tree tensor networks"

17/01/2023 11:30

First I report on recent numerical experiments with time-dependent
tree tensor network algorithms for the approximation of quantum spin
systems. I will then describe the basics in the design of time
integration methods that are robust to the usual presence of small
singular values, that have good structure-preserving properties (norm,
energy conservation or dissipation), and that allow...

10. Louise Gassot: "Zero-dispersion limit for the Benjamin-Ono equation on the torus"

Louise Gassot

17/01/2023 14:30

We discuss the zero-dispersion limit for the Benjamin-Ono equation on the torus given a bell-shaped initial data. We prove that the solutions admit a weak limit as the dispersion parameter tends to zero, which is explicit and constructed from the Burgers' equation. The approach relies on the complete integrability for the Benjamin-Ono equation from Gérard, Kappeler and Topalov, and also on the...

29. Hajer Bahouri: "On the global well-posedness for the derivative nonlinear Schrödinger equation on the real line"

Hajer Bahouri

17/01/2023 15:30

In this joint work with Galina Perelman, we have been interested in the question of global well-posedness for the derivative nonlinear cubic Schrödinger equation on the real line. This equation known as (DNLS) appeared in the 80's in the study of the one-dimensional compressible magneto-hydrodynamic equation in the presence of the Hall effec and the propagation of circular polarized...

12. Raffaele Carlone: "A ionization model"

17/01/2023 17:00

In the first part of the talk, it will be discussed the dynamics of a polaron (a quantum particle coupled to bosonic fields) in the quasi-classical regime: in such a regime, the effective dynamics for the quantum particles are approximated by the one generated by a time-dependent point interaction, i.e., a singular time-dependent perturbation of the Laplacian supported in a point.

In the...

13. Alessandro Teta: "Many-particle systems with contact interactions in dimension three"

18/01/2023 09:00

Hamiltonians with contact (or zero-range) interactions are useful models to analyze the behaviour of quantum systems at low energy in different contexts. In this talk we discuss the mathematical aspects of the construction of such Hamiltonians in dimension three as self-adjoint and lower bounded operators in the appropriate Hilbert space. We first consider the case of a system made of three...

14. Serena Cenatiempo: "A second order upper bound for the ground state energy of dilute Bose gases"

18/01/2023 10:30

Back in 2009 H.-T. Yau and J. Yin established a second order upper bound for the ground state energy of dilute Bose gases in the thermodynamic limit, finally in agreement with a elebrated prediction due to Bogoliubov and, in more explicit terms, by Lee, Huang and Yang. In this talk we describe recent ideas allowing us to establish the same result for a larger class of potentials (namely...

15. Benjamin Schlein: "Upper bounds on the ground state energy of dilute hard spheres"

18/01/2023 11:30

We review some recent estimates on the energy of bosons interacting through hard-sphere potentials. We first discuss Bose gases in the Gross-Pitaevskii regime, in which N hard spheres with radius of order 1/N move on the unit torus; in this setting, we show an upper bound for the ground state energy, valid up to errors that vanish as N tends to infinity. We conclude presenting a simple new...

20. Lucrezia Cossetti: "A limiting absorption principle for time-harmonic isotropic Maxwell and Dirac equation"

19/01/2023 10:30

In this talk we investigate the L^p − L^q mapping properties of the resolvent associated with the time-harmonic isotropic Maxwell and perturbed Dirac operator. As spectral parameters close to the spectrum are also covered by our analysis, we establish a L^p − L^q type limiting absorption principle for these operators. Our analysis relies on new results for Helmholtz systems with zero order...

21. Loïc Le Treust: "The Dirac bag model in strong magnetic field"

19/01/2023 11:25

We study Dirac operators on two-dimensional domains coupled to a magnetic field perpendicular to the plane. We focus on the infinite-mass boundary condition (also called MIT bag condition). In the case of bounded domains, we establish the asymptotic behavior of the low-lying (positive and negative) energies in the limit of strong magnetic field.

16. Liviu Ignat: "Asymptotic behavior of solutions for some diffusion problems on metric graphs"

19/01/2023 14:15

In this talk we present some recent result about the long time behavior of the solutions for some diffusion processes on a metric graph. We study evolution problems on a metric connected finite graph in which some of the edges have infinity length. We show that the asymptotic behaviour of the solutions of the heat equation (or even some nonlocal diffusion problems) is given by the solution...

17. Paolo Antonelli : "Stability of Cnoidal Waves for the Damped Nonlinear Schrödinger Equation"

19/01/2023 15:10

We consider the cubic nonlinear Schrödinger (NLS) equation with a linear damping on the one dimensional torus and we investigate the stability of some solitary wave profiles within the dissipative dynamics. The undamped cubic NLS equation is well known to admit a family of periodic waves given by Jacobi elliptic functions of cnoidal type. We show that the family of cnoidal waves is orbitally...

30. Cristian Cazacu: "Uncertainty principles: sharp constants, extremal functions and stability results"

19/01/2023 16:30

First of all, in this talk we focus on two well-known uncertainty principles applied to special classes of vector fields and we show that the sharp constants improve with respect to case of scalar fields whereas minimizers are described explicitly. These results are also extended to more general functional inequalities of Caffarelli-Kohn-Nirenberg type. Secondly, we provide optimal constants...

22. Pierangelo Marcati: "Existence and Stability of almost finite energy weak solutions to the Quantum Euler­ Maxwell system"

20/01/2023 09:00

We prove the existence of global in time, finite energy, weak solutions to a quantum magneto hydrodynamic system [7] (QMHD) with large data, modeling a charged quantum fluid interacting with a self generated electromagnetic field. The analysis of QMHD relies upon the use of Madelung transformations. The rigorous derivation requires non­trivial smoothing estimates, which are obtained by...

23. Luigi Barletti: "Functional calculus in phase-space and application to electron hydrodynamics in graphene"

20/01/2023 10:30

We present a systematic method to compute the formal semiclassical expansion of functional calculus in the framework of the phase-space representation of quantum mechanics. This is particularly useful to compute subleading corrections to the classical Maxwell-Boltzmann, Fermi-Dirac or other local equilibrium distributions. We also show how these results can be applied to derive quantum...

24. Vittorio Romano: "Analytical and simulation aspects of charge transport in graphene"

20/01/2023 11:30

The last years have witnessed a great interest for 2D-materials due to their promising applications. One of the most investigated is graphene which is considered as a potential new material to be exploited in nano-electronic and optoelectronic devices.
Charge transport in graphene can be described with several degrees of physical complexity [1]. At quantum level an accurate model is...

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19. Cristian Cazacu: "Uncertainty principles: sharp constants, extremal functions and stability results"

First of all, in this talk we focus on two well-known uncertainty principles applied to special classes of vector fields and we show that the sharp constants improve with respect to case of scalar fields whereas minimizers are described explicitly. These results are also extended to more general functional inequalities of Caffarelli-Kohn-Nirenberg type. Secondly, we provide optimal constants...

11. Perelman

Galina Perelman

18. Raffaele Carlone: "A ionization model"

In the first part of the talk, it will be discussed the dynamics of a polaron (a quantum particle coupled to bosonic fields) in the quasi-classical regime: in such a regime, the effective dynamics for the quantum particles are approximated by the one generated by a time-dependent point interaction, i.e., a singular time-dependent perturbation of the Laplacian supported in a point.

In the...