# Spectral Theory and pRObability in Mathematical physics (STROM)

Wednesday, June 8, 2022

at

IRMA, Université de Strasbourg

organized by

Nalini Anantharaman, Yohann Le Floch, Semyon Klevtsov, Martin Vogel, Xiaolin Zeng

This workshop focuses on Spectral Theory and Probability in Mathematical Physics, in particular on Random Matrices, Random Schrödinger Operators and Statistical mechanics.

Strasbourg is a lovely town in the northeastern part of France, located at the border with Germany; below you can find some pictures of the city.

Funding: We have additional funding possibilities to cover travel and hotel costs for interested participants. PostDoc and PhD students are particularly invited to apply for funding.

## Talks

• Mitia Duerinckx CNRS et Université Paris-Sud

• Cherenkov radiation and quantum friction

• This talk is devoted to several translation-invariant QFT models for a non-relativistic quantum particle interacting with a quantized relativistic field of bosons. We aim at the rigorous study of Cherenkov radiation or friction effects at weak coupling, which amounts to the metastability of the embedded mass shell of the free non-relativistic particle. Although the problem is naturally approached by Mourre’s commutator method, regularity issues are known to be inherent to QFT models and restrict the application of the method. We introduce a new, non-standard construction of Mourre conjugate operators, which differs from second quantization and allows to circumvent regularity issues, thus leading us to improve previous results on the topic. We also comment on the quantum diffusion conjecture, which can be rephrased as a similar resonance question for random Schrödinger operators that remains open.

• San Vũ Ngọc Université de Rennes 1

• Hamiltonians with Ak singularities and semiclassical inverse spectral theory
• This is an ongoing joint work with Nikolay Martynchuk. We try to classify 1D Hamiltonians with a singularity at the origin of the form $$\xi^2 + x^k$$, with the hope to perform inverse spectral theory: from the spectrum of a pseudodifferential operator, which we assume discrete, can you recover the classical Hamiltonian? The Morse case k=2 is well known, but the general case seems open. A natural idea is to transform the operator into a Schrödinger operator, and apply known results in this case. I will present first results concerning the analytic and $$C^\infty$$ classification.
• Wei-min Wang CNRS and Cergy Paris Université

• Anderson localization for the nonlinear random Schroedinger equations

• We review results on nonlinear Anderson localization. This talk is based on the joint works with J. Bourgain, and more recently with W. Liu.

• Daniel Sanchez-Mendoza Université de Strasbourg

• Principal Eigenvalue and Landscape Function of the Anderson Model on a Large Box.

• We state a precise formulation of a conjecture concerning the product of the principal eigenvalue and the sup-norm of the landscape function of the Anderson model restricted to a large box. We first provide the asymptotic of the principal eigenvalue as the size of the box grows and then use it to give a partial proof of the conjecture. For the one dimensional case we give a complete proof.