Graduate Seminar
Operator kernel estimates and their applications
Prof. M. Disertori
Time and place: Thursday 1618 Room N0.008
Tentative program
10.10.2019 Introduction
24.10.2019 Cauchy integral representation and Helffer Sjöstrand representation I
31.10.2019 CombesThomas estimate
14.11.2019 Applications
21.11.2019 HelfferSjöstrand representation II
28.11.2019 An application: decay of correlations
9.1.2020 Putting together what we did
16.1.2020 WittenLaplacian approach: Introduction
23.1.2010 WittenLaplacian: 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_{x}L^{1}δ_{y} , or more generally of δ_{x}f(L)δ_{y }where f is a bounded continuous function. When L is a selfadjoint 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 (Lz)^{1}. The price to pay is to work with complex valued functions. The two most important examples are Cauchy integral formula and HelfferSjöstrand representation. The first requires f to be analytic, the second holds for any smooth function. As a consequence decay properties of δ_{x}f(L)δ_{y }can be deduced from decay estimates for the resolvent operator (Lz)^{1}. A key tool in this context is the CombesThomas 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 C^{2 }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 socalled WittenLaplacian (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

JeanMarc Bouclet: 'An introduction to pseudodifferential 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

AizenmanWarzel: 'Random Operators', Graduate Studies in Mathematics AMS Vol 168

AizenmanElgartNabokoSchenkerStolz: ‘Moment analysis for localization in random Schrödingetr operators’ Invent. math. 163, 343–413 (2006)

GerminetKlein: 'Operator kernel estimates for functions of generalized Schrödinger operators' PAMS 131 (3), 911920 (2002)

BoucletGerminetKlein: ‘Subexponential 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 'Selfadjoint operators and solving the Schrödinger equation' tutorial 2014