Graduate Seminar 

Operator kernel estimates and their applications

Prof. M. Disertori

 

Time and place: Thursday 16-18 Room N0.008

Tentative program

10.10.2019  Introduction

24.10.2019 Cauchy integral representation and Helffer Sjöstrand representation I

31.10.2019  Combes-Thomas estimate

14.11.2019  Applications

21.11.2019 Helffer-Sjöstrand representation II

28.11.2019  An application: decay of correlations

9.1.2020    Putting together what we did

16.1.2020 Witten-Laplacian approach: Introduction

23.1.2010  Witten-Laplacian: alternative proof of HS II

 

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 order differential operator, R is a fixed matrix and 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. When L is a self-adjoint operator the function f(L) can be defined via spectral theorem:

                                                     f(L) =  ∫ f(x) dμ(x)

where dμ(x) is an operator valued measure. This measure is not always easy to handle.

In the fist part of the seminar we will consider alternative integral representations for f(L) that only involve Lebesque measures  and the operator resolvent (L-z)^-1. The price to pay is to work with complex valued  functions. The two most important examples are Cauchy integral formula and Helffer-Sjöstrand representation. The first requires f to be analytic, the second holds for any smooth function. As a consequence decay properties of δ_xf(L)δ_y can be deduced from decay estimates for the  resolvent operator (L-z)^-1. A key tool in this context is the  Combes-Thomas estimate.

In the second part of the seminar we will concentrate on a differential operator L arising naturally in statistical mechanical models. In this case we consider a  measure of the form

                                     dμ(φ) =dφ_Λ e^-Φ(∇φ) = dφ_Λ e^−F(∇φ) e^−M(φ)

where φ : Λ → H, with Λ ⊂⊂ Z^d  , H is a real or complex finite dimensional Hilbert space,  F,M  are C²  functions. The main problem is to study existence and properties of this measure in the limit  Λ → Z^d

In this context an important quantity is the correlation <(f-<f>)(g-<g>)> where

                                                     <f>:=   ∫ f(φ) dμ(φ).

A famous result due to Helffer and Sjöstrand relates the correlation above with the inverse of the differential operator acting on vector valued functions u: Λ → R^d as follows

                                                (Lu)_j =   -(Δu_j)(x)+  (Φ''(x)u(x))_j

This is the so-called Witten-Laplacian (or deformed Laplacian). Informations on the correlations can then be deduced by spectral properties of L.

Prerequisites. Functional analysis and Introduction to PDE. Some basic knowledge in statistical mechanics may be useful but is not necessary.

 

 

Some bibliography

 

• Jean-Marc Bouclet: 'An introduction to pseudo-differential operators'  

• Davies E.B.: 'Spectral Theory and Differential Operators' Cambridge University Press 1995

• Davies E.B.: 'Linear Operators and their spectra' Cambridge University Press 2007

• Aizenman-Warzel: 'Random Operators', Graduate Studies in Mathematics AMS Vol 168

• Aizenman-Elgart-Naboko-Schenker-Stolz: ‘Moment analysis for localization in random Schrödingetr operators’ Invent. math. 163, 343–413 (2006)

• Germinet-Klein: 'Operator kernel estimates for functions of generalized Schrödinger operators' PAMS 131 (3), 911920 (2002)  

• Bouclet-Germinet-Klein: ‘Sub-exponential decay of operator kernels for
  functions of generalized schrödinger operators’, Proceedings of the american mathematical society Volume 132, Number 9, Pages 2703–2712
  Article electronically published on April 21, 2004

• 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

• Helffer:  ‘Semiclassical Analysis, Witten Laplacians, and Statistical Mechanics’, Series on Partial Diffential Equations and Applications Vol 1, World Scientific

• Stolz 'Self-adjoint operators and solving the Schrödinger equation' tutorial 2014