Prof. M. Disertori, Dr. Rojas-Molina
The seminar will take place on Tuesdays 10-12, Room N 0.008
- October 24 "Introduction to spectral theory for unbounded operators"
- November 7 "Helffer-Sjöstrand representation I"
- November 14 "Combes-Thomas estimate"
- November 21 "Helffer-Sjöstrand representation II"
- December 5 "An application: decay of correlations"
- December 12 "Another application: homogeneization Part I"
- December 19 "Another application: homogeneization Part II"
Jean-Marc Bouclet: 'An introduction to pseudo-differential operators'
Aizenman-Warzel: 'Random Operators', Graduate Studies in Mathematics AMS Vol 168
Germinet-Klein: 'Operator kernel estimates for functions of generalized Schrödinger operators' PAMS 131 (3), 911920 (2002)
Naddaf and Spencer: On homogenization and scaling limit of some gradient perturbations of a massless free field. Comm. Math. Phys. 183 (1997), no.1, 5584
Description. A large number of problems in mathematical and theoretical physics can be reformulated in terms of a generalized Schrödinger equation of the form
i∂t ψ = Lψ,
where L = DRD + V, D is a first oder differential operator, R is a fixed matrix V a multiplication operator. Prominent examples include the standard Schrödinger equation and classical wave equation (like acoustic and Maxwell). A major problem is the study of the corresponding operator kernel δxL-1δy , or more generally of δxf(L)δy where f is a bounded continuous function.
In this seminar we will learn some fundamental tools to study these problems, namely Combes-Thomas estimate and Helffer-Sjöstrand representation. We will then see two applications.
(a) Study correlations for gradient type measures of the form
dμ(φ) = dφΛ e−F(∇φ)
where φ : Λ → H, with Λ ⊂⊂ Z d , H is a real or complex finite dimensional Hilbert space, F is C2 and convex, and Λ → Zd
(b) Prove decay estimates of the form |δxf (L)δy | ≤ g(|x − y|), where g(u)→0 as u→∞.
Prerequisites. Functional analysis and Introduction to PDE. Some basic knowledge in statistical mechanics may be useful but is not necessary.