## Graduate Seminar S4B2

**Operator kernel estimates and their applications**

Prof. M. Disertori

Time and place: to fix

If you are interested in the seminar, please send me an email by March 7.

When I will know the approximate number of participants I will prepare a precise list of possible topics and a doodle to fix a time/date.

A tentative list of titles (that may be enlarged or reduced according to the number of participants) is given below.

**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 has the form 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 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_{x}f(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) = e^{-Φ(∇φ)} dφ Φ(∇φ)= F(∇φ)+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 so-called Witten-Laplacian (or deformed Laplacian). Informations on the correlations can then be deduced by spectral properties of L.

**Prerequisites**. Functional analysis. Some basic knowledge in statistical mechanics may be useful but is not necessary.

**Preliminary list of topics. **

I will give in any case a first talk summarizing the notions that you will need on spectral calculus

**1.*** Cauchy integral formula and Helffer-Sjöstrand representation I*

HS formula 1 [D] pages 24 and 25 and [B] sect. 4.3

Cauchy representation eq 13.38 [AW page 210]

**2.** *Combes-Thomas estimate*

proof in the discrete case [AW] Section 10.3

**3.** *Example of application: two point function for electrons in a disordered media *

[AW] Section 13.4 (proof by Cauchy formula)

[AENSS] statement of Theorem 1.1 and Application (3), Remark (11) in Appendix (A)

**4.** *Helffer-Sjöstrand representation II*

[NS] Section 1.1 (the model) only until end of page 2 (without Brascamp-Lieb)

Section 1.3 (sketch of the H-S construction (only in finite volume) until (eq 1.10)

proof of Brascamp-Lieb from eq 1.10 Section 2.1 (invertibility of the operator L)

**5.*** An application: decay of correlation*s

[NS] construction of the infinite volume measure (page 3)+statement of Theorems B,

Section 2.2 proof of Theorems B and C

**6.** *Witten-Laplacian approach. Introduction*

[H] Section 2.1-2.5

**7. ***Witten-Laplacian approach: alternative proof of HSII*

[H] Section 2.6

**8.*** Witten-Laplacian approach: decay of correlations for logconcave measures.*

[H] Selected parts of Section 8

**Bibliography**

[AW] Aizenman-Warzel: '*Random Operators*', Graduate Studies in Mathematics AMS Vol 168.

[B] Jean-Marc Bouclet: '*An introduction to pseudo-differential operators*'

www.math.univ-toulouse.fr/~bouclet/Notes-de-cours-exo-exam/M2/cours-2012.pdf

[D] Davies: ‘*Spectral **t**heory and differential operators*’

[D1 ] Davies: *'Linear Operators and their spectra*' Cambridge University Press 2007

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

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

[BKG] 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

[NS] 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

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

[S] Stolz 'Self-adjoint operators and solving the Schrödinger equation' tutorial 2014