Prof. M. Disertori, Dr. Rojas-Molina

 The seminar will take place on  Tuesdays 10-12, Room N 0.008

 

 

Program 

• 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"

 

 

Bibliography

 

• 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 δ_xL^-1δ_y , or more generally of δ_xf(L)δ_y 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) = dφ_Λ 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 → Z^d

(b) Prove decay estimates of the form |δ_xf (L)δ_y | ≤ 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.