On the Schrodinger operator with white noise potential

Cyril Labbé

This talk will focus on the random Schrodinger operator on R obtained by perturbing the Laplacian with a white noise. First, I will present an Anderson localization result for this operator: the spectrum is pure point and the eigenfunctions exponentially localized. Second, I will present various results on the eigenvalues and eigenfunctions of the finite-volume approximation of this operator in the infinite volume limit: it turns out that many different behaviors can be observed according to the energy regime one focuses on. This is based on a series of joint works with Laure Dumaz.

Slides

TBA

Ilya Chevyrev

TBA

TBA

Fabio Toninelli

TBA

TBA

Filippo Nava

TBA

The damping of phonons in Bose gas at low temperature

Lorenzo Pettinari

Condensed Bose gases can be effectively described in terms of quasi-particles, commonly referred to as phonons. Their dynamics is captured by a c-number condensate Hamiltonian consisting of a quadratic term supplemented by third- and fourth-order perturbative corrections. These additional interaction terms render the phonons unstable, giving rise to two distinct decay processes known as Beliaev and Landau damping. From a mathematical perspective, such decay mechanisms should manifest as a broadening of the Bogoliubov dispersion relation in the thermodynamic limit. 

To validate this picture, I will present two different approaches to deriving the phonon decay rates. The first is inspired by the W ^*-algebraic framework of Jakšić–Pillet, employing Standard Representations and perturbative expansions of a suitably chosen vector state. The second method is based on the analysis of two-body correlation functions. Both approaches yield the same imaginary correction to the Bogoliubov dispersion relation, which in turn determines the expected broadening. Furthermore, our approaches offer a new perspective on the decay of phonons in terms of the left and right components of these quasi-particles. 

The talk is based on joint work with Jan Dereziński and may be viewed as a modern elaboration of the classical contributions of Beliaev, Hohenberg–Martin, and others.

Slides

Equilibrium states for non relativistic Bosonic QFT

Stefano Galanda

In this talk we present the construction of equilibrium states at positive temperature, with and without Bose-Einstein condensation, for a non-relativistic Bosonic QFT (gas of Bose particles) on $\mathbb{R}^3$, interacting through a localised two body interaction. We use methods of quantum field theory in the algebraic formulation and of quantum statistical mechanics in the operator algebraic setting to obtain this result. Moreover, in order to prove convergence of the correlation functions, we rearrange the perturbative series introducing an auxiliary stochastic Gaussian field which mediates the interaction of the Bosonic field. Limits where the localisation of the two-body interaction is removed are eventually discussed in combination with other regimes. This talk is based on a collaboration with Nicola Pinamonti.

Slides

TBA

Harprit Singh

TBA

TBA

Nicolò Drago

TBA

TBA

Michele Schiavina

TBA

TBA

Lorenzo Zambotti

TBA

TBA

Francesco Caravenna

TBA

TBA

Alessio Ranallo

TBA

The semigroup of an Itô diffusion as an exotic B-series

Alberto Bonicelli

In this talk, after justifying the expansion of the semigroup of a one-dimensional Itô diffusion as a power series in time, I will build on previous results on expansions labelled by exotic rooted trees to derive an explicit expression for the combinatorial factors involved. A key step is the extension of the notion of tree factorial and Connes-Moscovici weights to this richer family of rooted trees. As a result, we obtain an exotic Butcher series representation of the semigroup, suitable for a comparison with the perturbative path integral construction of the statistics of the diffusion, known in the literature as Martin-Siggia-Rose formalism. Computations in the latter framework are based on the erroneous assumption that the measure of the path integral can be seen as a perturbation of a Gaussian measure. Resorting to multi-indices to represent pre-Feynman diagrams, I will shed some light on why, even if starting from such an assumption, the results are correct.

Slides