Oberseminar "Analysis and Mathematical Physics"

Summer Term 2020

Abstracts

Phan Thanh Nam: Correlation energy of a weakly interacting Fermi gas

In theoretical physics, corrections to the Hartree-Fock approximation are often calculated using perturbation methods. However, showing convergence of perturbative expansions is a serious problem. In the talk, I will discuss a non-perturbative approach to derive rigorously the correlation energy for a weakly interacting Fermi gas in the mean-field regime. Our result agrees with the prediction from the random-phase approximation. This is joint work with Niels Benedikter, Marcello Porta, Benjamin Schlein, and Robert Seiringer.




Arnaud Triay: The Scott correction in Dirac-Fock theory

The Scott correction is the second order in the expansion of the ground state energy of large atoms. It originates from electrons interacting very closely to the singular potential generated by the nucleus. Because these electrons have very high momenta, a physically relevant theory requires to treat them relativistically. In quantum mechanics the kinetic energy of the relativistic electron is given by the Dirac operator which models both the electron, with positive energy, and its antiparticle, the positron, with negative energy. Because the spectrum of the Dirac operator is unbounded below, the many body Dirac operator is ill-defined and it is even unclear what definition for a ground state to take. For this reason, one usually considers projected Dirac operators, onto either the positive sub-space of the free Dirac operators or Dirac with some potential. In the Dirac-Fock theory, the set of admissible states satisfy the self-consistent constraint to be orthogonal to the sub-space of negative energy of their own the mean-field operator. We are interested to the derivation of the Scott correction in this setting.




Robert Seiringer: Emergence of Haldane pseudo-potentials in systems with short-range interactions

We present a derivation of Haldane?s pseudo-potential operators for many-body quantum systems with short-range interactions subject to a (homogeneous) magnetic field. In particular, we show that Laughlin?s wavefunctions emerge as limits of ground states of such models under suitable confinement.




Nikolai Leopold: Landau-Pekar equations and quantum fluctuations for the dynamics of a strongly coupled polaron

The Fröhlich model describes the interaction between an electron and the quantized phonons of a ionic crystal. In this talk, I will discuss its time evolution in the limit of large coupling constant ?. For initial data of Pekar product form with coherent phonon field and with the electron minimizing the corresponding energy, I will present a norm approximation of the evolution, valid up to times of order ?^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. This is joint work with David Mitrouskas, Simone Rademacher, Benjamin Schlein, and Robert Seiringer.




Cambyse Rouzé: Discriminating unitary channels with energy-constrained states.

TThe task of distinguishing unknown objects is arguably a fundamental one in experimental science. Quantum state discrimination constitutes one of the simplest examples of a problem of this sort. More generally, the question of distinguishing between quantum channels, when given access to an ensemble of input states and measurements characterised by some physical constraints (entanglement, energy,...) has gained a central role in the flourishing field of quantum information science. In this talk, I will investigate the energy-constrained (EC) diamond norm distance between unitary channels acting on possibly infinite-dimensional quantum systems, and establish a number of new results. I will first prove that optimal EC discrimination between two unitary channels does not require the use of any entanglement. Extending a result by Acín, I will also show that a finite number of parallel queries suffices to achieve zero error discrimination even in this EC setting. In the second part of the talk, I will apply these results to two situations of practical relevance: first, I will derive bounds on a novel type of quantum speed limits, which apply to pairs of quantum dynamical semigroups. Then, I will establish a Gaussian version of the Solovay–Kitaev algorithm for the efficient approximation of unitary gates on a quantum circuit, with the approximation error being measured with respect to the EC diamond norm relative to the photon number Hamiltonian. This talk is based on the preprint arXiv:2006.06659, which is joint work with Simon Becker, Nilanjana Datta and Ludovico Lami.




Andrea Posilicano: On the self-adjointness of H+A*+A

Let H and A be linear operators in a Hilbert space, H self-adjoint (playing the role of the generator of the free dynamics) and A H-bounded (playing the role of the annihilator operator). By a twofold application of a Krein-type formula, we provide self-adjoint realizations H^ of the formal Hamiltonian H+A*+A, a formula for the resolvent difference (-H^+z)-1 - (-H+z)-1 and the explicit characterization of the operator domain D(H^). We also consider the problem of the description of H^ as the norm resolvent limit of sequences H+An*+An+En, where the An's are regularized operators approximating A and the En's are suitable renormalizing bounded operators. These results show the connection between the construction of singular perturbations of self-adjoint operators by Krein's resolvent formula and the nonperturbative theory of renormalizable models in Quantum Field Theory; in particular, as an explicit example, we consider the Nelson model.




Tadahiro Miyao: Positivity improving property of the Nelson semigroup

The Nelson model is a toy model describing the interaction between spinless nucleons and a scalar meson field. Because the mathematical structure of the model is characteristic, it has been actively studied by mathematical physicists and mathematicians. In this talk, I will briefly explain how we prove that the semigroup generated by the Nelson Hamiltonian improves the positivity in the Fock representation. Some applications and extensions will be discussed.




Vedran Sohinger: Gibbs measures of nonlinear Schrödinger equations as limits of many-body quantum states

Gibbs measures of nonlinear Schrödinger equations are a fundamental object used to study low-regularity solutions with random initial data. In the dispersive PDE community, this point of view was pioneered by Bourgain in the 1990s. The nonlinear Schrödinger equation can be viewed as a classical limit of many-body quantum theory. We are interested in the problem of the derivation of Gibbs measures as mean-field limits of Gibbs states in many-body quantum mechanics. Our proof is based on a functional integral representation of the quantum Gibbs state. In this framework, the limit is formally deduced by using a infinite-dimensional stationary phase argument. In order to make this rigorous, we introduce an auxiliary white-noise field, through which the functional integral is represented in terms of propagators of heat equations with time-dependent periodic random potentials, and subsequently as a gas of interacting Brownian paths. This is joint work with J. Fröhlich, A. Knowles, and B. Schlein