• Tingueu el micròfon apagat tota l'estona excepte durant el "Cafè i galetes".
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• Si voleu fer una pregunta, si us plau demaneu torn a través del xat i espereu que els moderadors us donin la paraula.
• Si és possible, entreu amb antelació. Els que no useu un compte de correu de la UPC haureu d'esperar a ser admesos per accedir-hi. (Això aplica a les sessions Google Meet de la UPC).

Dia: Dimecres, 18 de maig de 2022

Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.

També ONLINE https://meet.google.com/zvg-pajn-owr

• Hora: 16h00m

A càrrec de: Dario Bambusi (Università degli studi di Milano)

Títol: Growth of Sobolev norms for unbounded perturbations of the Laplacian on flat tori (towards a quantum Nekhoroshev theorem)

Resum: I will present a study of the time dependent Schrödinger equation -i∂Ψ/∂t=-ΔΨ+v(t,x,-i∇)Ψ on a flat d dimensional torus. Here V is a time dependent pseudodifferential operator of order strictly smaller than 2. The main result I will give is an estimate ensuring that the Sobolev norms of the solutions are bounded by tǫ. The proof is a quantization of the proof of the Nekhoroshev theorem, both analytic and geometric parts. Previous results of this kind were limited either to the case of bounded perturbations of the Laplacian or to quantization of systems with a trivial geometry of the resonances, lik harmonic oscillators or 1-d systems. In this seminar I will present the result and the main ideas of the proof.

• Hora: 17h15m

A càrrec de: Beatrice Langella (SISSA, Trieste)

Títol: Growth of Sobolev norms in quasi integrable quantum systems

Resum: In this talk I will analyze an abstract linear time dependent Schrödinger equation of the form -i∂Ψ/∂t=(H₀+V(t))Ψ (1) with H₀ a pseudo-differential operator of order d > 1 and V(t) a time dependent family of pseudo-differential operators of order strictly less than d. I will introduce abstract assumptions on H₀, namely steepness and global quantum integrability, under which we can prove a |t|ε upper bound on the growth of Sobolev norms of all the solutions of (1). The result I will present applies to several models, as perturbations of the quantum anharmonic oscillator in dimension 2, and perturbations of the Laplacian on a manifold with integrable geodesic flow, and in particular: flat tori, Zoll manifolds, rotation invariant surfaces and Lie groups. The case of several particles on a Zoll manifold, a torus or a Lie group is also covered. The proof is based a on quantum version of the proof of the classical Nekhoroshev theorem.

This is a joint work with Dario Bambusi.

Last updated: Mon May 16 16:07:46 2022