Series on Partial Differential Equations and Applications - V o l . 1
7
Bernard
Helffer
classical Analysis, en Laplacians, and Statistical
World Scientific
Mechanics
Semiclassical A n a l y s i s , W i t t e n Laplacians, and Statistical
Mechanics
Series on Partial Differential Equations and Applications Editorial Board Editor-in-Chief Fanghua Lin Courant Institute of Mathematical Science New York University New York, NY 10012-1110 USA Email:
[email protected]
Editors Lawrence C. Evans Department of Mathematics University of California Berkeley, CA, 94720-3840 USA Email:
[email protected]
Chang Shou Lin National Chung Cheng University Department of Mathematics Ming-hsiung, Chia Yi, 621 Taiwan Email:
[email protected]
Paul Yang Princeton University Mathematics Department Fine Hall, Washington Road Princeton NJ 08544-1000 USA Email:
[email protected]
Ma Li Department of Mathematics Tshinghua University Beijing 100084 China
Pan Xingbin Department of Mathematics National University of Singapore Singapore 119260 Email:
[email protected] Xu Xingwang Department of Mathematics National University of Singapore Singapore 119260 Email:
[email protected]
Neil Trudinger School of Mathematical Sciences Australian National University Canberra, ACT 0200 Australia Email:
[email protected] Y. Giga Department of Mathematics Hokkaido University Sapporo 060-0810 Japan Email:
[email protected]
Series on Partial Differential Equations and Applications - V o l . 1
Semiclassical Analysis, W i t t e n Laplacians, and Statistical
Mechanics
Bernard Helffer Universite Paris Sud, France
U J j World Scientific Vw
New Jersey London •Singapore* Singapore • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
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Preface
In this book we shall analyze with techniques coming mainly from partial differential equations (PDE) and of semi-classical analysis problems coming from statistical mechanics. Our main object of analysis is a (family of) measure(s) representing the probability of presence of m particles in interaction and having the form d ^
:= Z(m, h)-1 exp
- ^
dX
(m € IN) where • Z(m, h) is a normalization constant, .
$(m)
is a C°° function defined on JR m , tending to oo at oo, with a specific structure coming from statistical mechanics (usually a perturbation of Y^?=i ^i.xi) taking account of the interaction between nearest neighbours), • ft is a strictly positive parameter playing the role of an effective planck constant, • dX is the Lebesgue measure on JRm • the integer m represents the cardinal of a set A in the lattice ZZ which will tend infinity. We have consequently two main parameters h and m. The limit h —> 0 corresponds to the so called semi-classical limit h (which can actually corresponds to the temperature) and m —>• +00 corresponds to the so called thermodynamic limit, when a large number of particles is involved. Our aim in this book is to explain, after presentation of the physical background, two techniques for analyzing these problems.
VI
Preface
The first one is the so called transfer matrix approach. In this method which takes its origin in the analysis of the Ising model, the thermodynamic limit problem is solved by reduction to the analysis of a compact operator on a one-particle Hilbert space. We shall see how in this context the semi-classical mechanics can be a useful tool in the analysis of the spectral properties of this operator. The second one is the technique of the Witten Laplacian approach which gives a new light and suggests new proofs for the analysis of Poincare estimates and log-Sobolev estimates in relation with the measure of the decay of correlations. The main difficulty will be to control constants or remainders independently of the dimension. We shall show in this context how techniques coming from the theory of partial differential equations and applied to the Witten Laplacian on one-forms can be efficient for showing the uniformity of some constants. Although the problems we are treating are strongly motivated by statistical mechanics where probability theory plays an important role, we do not assume that the reader has a strong background in this theory. We will try instead to present all the material which is needed in spectral theory, semi-classical analysis and statistical mechanics in a rather selfcontained way. Let us add that we do not pretend at exhaustivity. We have preferred to present a walk through the theory. The initial version of this manuscript was prepared when teaching in Orsay and Toulouse in spring and winter 1998. But we include also, particularly in the notes, references to more recent contributions. First of all we would like to thank J. Sjostrand who has collaborated with us on many results presented here and T. Bodineau who was the first to push us to look at these log-Sobolev estimates and to bring us his knowledge in statistical mechanics in a fruitful collaboration. For various reasons (collaborations, invitations, fruitful discussions, comments or remarks), we would also like to thank V. Bach, D. Bakry, J.D. Deuschel, T. Jecko, M. Ledoux, A. Martinez, J.-M. Roquejoffre, T. Ramond, N. Yoshida, B. Zegarlinski and all the students or colleagues who have followed or discussed with us the various versions of the material presented here. We thank also A. Bardot for his help when preparing the final version of the manuscript. Finally, we acknowleddge the support of CNRS and of the European Union TMR grant FMRX-CT 96-0001.
Contents
Preface
v
Chapter 1 Introduction 1.1 Laplace integrals 1.2 The problems in statistical mechanics 1.3 Semi-classical analysis and transfer operators 1.4 About the contents
1 1 3 6 7
Chapter 2 W i t t e n Laplacians approach 2.1 De Rham Complex 2.2 Witten Complex 2.3 Witten Laplacians 2.4 Semi-classical considerations 2.5 An alternative point of view : Dirichlet forms 2.6 A nice formula for the covariance 2.7 Notes
9 9 13 14 15 16 17 21
Chapter 3 3.1 3.2 3.3 3.4
Problems in statistical mechanics with discrete spins The Curie-Weiss model The 1-d Ising model The 2-d Ising model Notes
Chapter 4 Laplace integrals and transfer operators 4.1 Introduction vii
23 23 25 28 29 31 31
viii
Contents
4.2
Classical Laplace method 4.2.1 Standard results 4.2.2 Transition between the convex case and the double well case 4.3 The method of transfer operators 4.4 Elementary properties of operators with integral kernels . . . . 4.5 Elementary properties of the transfer operator 4.6 Operators with strictly positive kernel and application 4.7 Thermodynamic limit 4.8 Mean value 4.9 Pair correlation 4.10 2-dimensional lattices 4.11 Notes Semi-classical analysis for the transfer operators 5.1 Introduction 5.2 Explicit computations for the harmonic Kac operator 5.3 Harmonic approximation for the transfer operator 5.4 WKB constructions for the transfer operator 5.5 The case of the Schrodinger operator in dimension 1 5.6 Harmonic approximation for the transfer operator: upper bounds 5.7 First conclusions about the splitting 5.8 Some elements about the decay 5.9 Splitting revisited 5.9.1 Preliminary discussion 5.9.2 Comparison of various problems 5.9.3 Upper bound of the splitting 5.10 Notes
31 31 33 34 35 39 40 42 43 44 45 49
Chapter 5
65 67 68 70 70 70 71 73
Chapter 6 6.1 6.2 6.3 6.4 6.5
Basic facts in spectral theory and on the Schrodinger operator Introduction Selfadjoint operators, spectrum and spectral decomposition . . Discrete spectrum, essential spectrum Essentially selfadjoint operators Examples
51 51 51 55 59 63
77 77 77 83 86 88
Contents
6.6 6.7 6.8 6.9
6.5.1 The free Laplacian 6.5.2 The harmonic oscillator More on selfadjointness The max-min principle Compactness Notes
ix
88 89 91 93 94 97
Chapter 7 Log-Sobolev inequalities 7.1 Introduction 7.2 Log-Sobolev inequalities in the strictly convex case 7.3 Around Herbst's argument : necessary conditions for log-Sobolev inequalities 7.4 Extension of the Bakry-Emery argument : convexity at infinity 7.5 The case of the circle 7.6 The case of the line 7.7 General remarks 7.8 Notes
101 101 103
Chapter 8 Uniform decay of correlations 8.1 Introduction 8.2 Lower bound for the spectrum of the Witten Laplacian . . . . 8.3 Uniform estimates for a family of 1-dimensional Witten Laplacians 8.4 A proof of the decay of correlations 8.5 Generalized Brascamp-Lieb inequality 8.6 Notes
133 133 136
115 118 120 123 128 130
139 142 147 148
Chapter 9 Uniform log-Sobolev inequalities 151 9.1 Introduction and preliminaries 151 9.2 Some log-Sobolev inequality for effective single spin phase . . . 152 9.3 The role of the decay estimates for log-Sobolev inequality . . . 155 9.4 Second part of the proof of the log-Sobolev inequality 158 9.5 Conclusion 166 9.6 Notes 167 Bibliography
169
Index
177
Chapter 1
Introduction
1.1
Laplace integrals
As mentioned in the preface, our basic object will be a measure on Mm : exp — $(X) dX. Here 3> will be a real C°° function which tends sufficiently rapidly to +00 as \X\ —>• +00. Typically, we can think of $(X) = a\X\2 with a > 0, but we would like of course to analyze more general cases. In order to give a sense to what follows let us assume that *(X)>±\X\*-C,
(1.1.1)
for some C > 0. In particular, under this assumption, the quantity Z=
f
exp-${X)dX
,
(1.1.2)
is finite and we can associate to the measure exp —$(X) dX a probability measure fi: dfi = Z-1exp-$(X)dX.
(1.1.3)
Associated to this measure, we define for a function / with polynomial growth the mean value of / by (/) = Z-1 [ JM™
f(X) exp - $ ( X ) dX = f fdfi. J 1
(1.1.4)
2
Introduction
For two functions / and g, we then define the covariance of two functions with polynomial growth / and g by Cov(/,<7) = < ( / - < / » ( < ? - ( < ? » > •
(1-1.5)
As simple example, let us consider :
$(X)=^2Aijxi-xj
,
(1.1.6)
where Aij is an m x m matrix which is symmetic and positive definite. The computation of the covariance of / and g with f(X) = Xk and g(X) = xi is given by {A'^u. For this given measure, we define also the variance by v a r / = Cov ( / , / ) .
(1.1.7)
Our main questions in this direction are to find natural assumptions permitting to prove the existence of the following inequalities called (P) (for Poincare) and (LS) (for logarithmic Sobolev or more shortly log-Sobolev). The first one is the existence of a constant C such that, for all / , (P)
var(/)
(1.1.8)
This inequality is called the Poincare inequality in reference to the known inequality relating ||/||i2(n) and || | V / | || for / 6 CQ°(Q), in the case when Jl is bounded ll/||L*(n)
(1.1.9)
The second one, which is of importance in statistical mechanics, is the so-called log-Sobolev inequality, stating the existence of a constant C such that, for any strictly positive / , we have (LS)
(/ln/)-(/)ln(/)<2C(|V/*|2).
(1.1.10)
In these two cases, the problem is not only to show the existence of the inequality for some constant C but also to be able to have an information on the smallest constant C such that this is true say Cp in the first case and CLS in the second case. Remark 1.1.1 It is interesting to replace Mm by a more general manifold M compact or
The problems in statistical
mechanics
3
not compact. Typically M = (Sk)N is quite interesting. The case k = 0 can be thought as the set {—1,+1} with two elements. In this case we have to consider discrete measures. More precisely, the Lebesgue measure on IRm is replaced by the measure on {—1, + l } m : (|(<5-i + <$+i)) <8> • • • ® (|(J_!+J+1)). Remark 1.1.2 For specific phases that will be described in the next section, it is important to analyze the above questions with a control of the constants with respect to the dimension or to other parameters.
1.2
The problems in statistical mechanics
In the context of statistical mechanics, our aim is to analyze for example the (family of) measure(s) exp — $A'f(X) dX in the case when $A,-f, which is associated with subsets A C Zd, has the form, for X € (MN)A,
where • X =
(XJ)J€AL;
• <j> is a one-particle phase on M with at least quadratic increase at oo; • j ~ k means that j and k are nearest-neighbors for the i1- distance in^; • the sum in (1.2.1) is over the non-oriented pairs of A x A. This model is also called the free boundary model. We can also consider the so-called periodic model where A is a cube A := A p e r which is identified to (Z/LZ)d. In this case the notion of nearest neighbors is changed accordingly. For example if d = 1, then we identify {1, • • •, L} with (Z/LZ) and L is considered as nearest neighbor of 1. The phase is then denoted by $A>Per. We shall also analyze in Chapters 8 and 9 models with other boundary conditions.
4
Introduction
Our main problem will be to analyze the properties of the measure dfiAj := exp -$A'f{X)
exp - $ A ' ' (X) dX)
dX/ I [
,
(1.2.2)
or of the measure dnA := exp -$ A -P e r (X) dX/ [ [
exp -$A
N
dX ) .
\J(R )*
(1.2.3)
J
We shall in particular analyze the covariance associating to / , g £ C£mp((MN)A) CovAJ(f,g)
= ( ( / - (f)AJ){g
-
<<7)A,/)>A,/
(1-2.4)
where ( • )\j denotes the mean value with respect to the measure d/j,Aj and C^mp{{JR ) A ) is the space of C°° functions with polynomial growth. More precisely, our main problems could be the following. Study of the asymptotic behavior of the Laplace integral f In / exp (~ |A| L'
_1
$AJ(X) , ) dX)
(1.2.5)
as |A| tends to oo, • Analysis of the behavior for i and j in A of the pair correlation : CorA'f(i,j):=
CovAJ{xi,Xj)
.
(1.2.6)
as d(i,j) is large (this takes a meaning as A "tends" to 2Z that is for example if A = [—L, • • •, L]d and* L is much larger than d(i, j)). The first problem will only be considered in the case when d = 1 through the technique of the transfer operator. The second question will be treated by the Witten Laplacian technique (for any d) as will be explained in the next chapter and by the transfer operator when d = 1 as will be sketched in the next section and analyzed in Chapters 4 and 5. It is clear that when there is no interaction, that is J = 0, then we have CovAJ{xi,xj)=0, *More precisely, one first takes i and j in Zd belong to A.
(1.2.7)
and then take L large such that i and j
The problems in statistical
mechanics
5
when i / j , that is in other words (xi-xj)
= (xi)(xj).
(1.2.8)
So it is natural to ask if something remains of this property as J is small. Outside examples with discrete spins which will be recalled in the next section, our main toy model will be a model coming from the field theory
V{m\X) = \ J > f c - xk+1)2 + b- JT>fc - y)\xk + yf
(1.2.9)
fc=i fe=i
where a > 0, b > 0 and y > 0. This corresponds to the case
d=l.
Although this will not be the main point in this book, it is useful to have some discussion about decay of correlations and phase transition. We can for example chose to analyze as A is large the expression aA:=
var^-t-^-) '
(1.2.10)
' j'eA
When $ is even with respect t o I i 4 —X, we observe that (XJ) = 0 and we get, in the case when A is considered as periodic *A:=j^rX)CorA(M)
(1-2.11)
When we have uniform decay of the correlations like | CorA(i,j)\
< Cexp-KdA(i,j)
,
(1.2.12)
with C and K > 0 independent of A, we immediately see that lim <7A = 0 .
(1.2.13)
A->Z d
This is typically the case when the interaction is small or when the phase is convex. Indeed, if we consider the case when
is strictly convex : <j>"(x) > p > 0 ,
(1.2.14)
it is easy to see that X t-» $ A ' / " 7 ( X ) satisfies, if J > 0, the condition Hess $*•'*•* > p .
(1.2.15)
6
Introduction
We will show that this implies an uniform Poincare inequality, When applying this inequality to the function X *-> A £3,- €A Xj, one can immediately obtain : aA < - ^ .
(1.2.16)
So we have obtained a proof of (1.2.13) when <j> is strictly convex which is valid for any J > 0. On the other hand, there are cases where this limit is non zero. This will be typically the case when d > 2, the phase
1.3
Semi-classical analysis and transfer operators
We consider $h = j ; and the question is now to understand what is going on as h —¥ 0. More precisely we have to analyze asymptotically expressions like f a(X) exp h~dX for a real phase <&. This leads to the research of the minima of 3> and then to the analysis of the quadratic approximation of $ near these minima. Here the Laplace method is used (which is directly related to the stationary phase theorem (and is actually more simple). In the context of statistical mechanics, the simplest model corresponds when d = 1 (see the previous section) to the analysis of : m
with xm+i
=x\.
m
Under suitable assumptions on (/>, the quantity :
ZJr):=Jexp-^m\X)dX, can be reinterpreted as the trace of the operator (Kh) m , where Kh is the operator whose distribution kernel is given by : Kh(x,y)
1 J 1 = exj>-—(t>(x) e x p - — \x - y\2 exp-—4>{y) .
About the
contents
7
A s m - > +00, it can be seen that the thermodynamic limit defined as : limm_>.+oo ^ In Z™ ls t n e logarithm of the largest eigenvalue of Kh. So the analysis of the thermodynamic limit is replaced by the spectral analysis of the operator Kh which will be called the transfer operator. We will describe in this book how to analyze this problem by the techniques of semi-classical analysis. 1.4
About the contents
Chapter 2 is devoted to a short presentation of the Witten Laplacian techniques. In Chapter 3 we recall how problems arise in statistical mechanics with discrete spins. Chapter 4 presents the Laplace method in connection with the analysis of the transfer operator. Chapter 5 shows how semiclassical analysis can be used for the spectral analysis of the transfer operator. This corresponds to a program suggested by M. Kac. Chapter 6 recalls some rather standard material in spectral theory of self adjoint operators but we take as basic examples of application the Witten Laplacians. Starting of Chapter 7, we begin to treat the logarithmic Sobolev inequalities. The book [DeStl] was a basic starting point for the presentation of log-Sobolev inequalities (which is rather complete in the case of compact manifols) but we were also influenced by notes of Ledoux [Ledl] and Antoniouk [AAl]. Chapter 8 is devoted to the proof of decay estimates with techniques initiated in collaboration with J. Sjostrand. Finally in Chapter 9, we present the proof of Zegarlinski's theorem [Ze] based on [Yoshl] and [BodHel]. A few words on the structure of the manuscript. At the end of each chapter, we have tried to give historical comments or references : This means that they are not usually inserted in the text and that the reader has to look at these comments spontaneously.
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Chapter 2
Witten Laplacians approach
2.1
D e Rham Complex
We denote* by fi°(iRm) the space of the real C°° 0-forms corresponding consequently to C°°(iR m ; M). We denote by ^{M"1) the space of the real C°° 1-forms corresponding consequently to C°°(Mm ; Mm) : m
(^ = Yl^dxJ
(2-1-1)
•
.7=1
More generally we can define p-forms and we recall that the m-forms can be identified with functions through the correspondence <j> >-» <j>dx\ A dxi • • • A dxm. The operation A associates to two forms w £ fip and a/ e fip the exterior product u A w ' in ilp+p and we have the property WAW' = ( - 1 )
W
(2.1.2)
VAW,
with in particular dx{ A dxi = 0 , dxi A dxj = —dxj A dxi .
(2.1.3)
We now recall that the exterior differential d is defined on with the following properties : • when restricted to fi°(2Rm), d is denoted by d^
(B'JLQW
(Mm)
and goes from
*We suppose that the reader has some familiarity with the p-forms and recall simply elementary facts in the case of Rm. The main interest of the theory is of course to extend this invariantly to general riemannian manifolds. 9
10
Witten Laplacians
Cl0(IRm) into n^R™)
approach
:
d<°)u = ^2(dXju)dXj .
(2.1.4)
3
More generally the restriction of d to £lp(Mm), d is denoted by d^ and goes from Qp(Mm) into ftP+1(iRm) • dod = 0,
(2.1.5)
that is d(p+i) 0 d(p)
= o , Vp = 0, • • •, m - 1 .
(2.1.6)
• When w e fip and a/ e ftp', then diP+p'){wAuj') = dF>u A J + ( - l ) p w A d ( p ' V in np+p'+\ In particular d^ on n x (iR m ) w = X)j=i Wjdij by
(2.1.7)
is the operator defined
dt1>w = Ei(d ( 0 ) w J -)Adx i = T.j,k(dxk^i)dxk A da:.,5 = Ej
rf(m) = 0 .
for
(2.1.8)
(2.1.9)
The De Rham Complex is then defined by dW
d« 1
o (->• n ° (-»• fi
M- n 2 (-»•••. M- n m M- o
(2.1.10)
and we can see this is a complex from (2.1.6). As it is classical, all this can be more generally defined on general C°° manifolds M. We are now interested in the introduction of a prehilbertian structure on these C°°-forms which are now assumed to be with compact support (in this case, we write QQ). This will permit to define by completion the corresponding Hilbert spaces of I? p-forms. It is sufficient for this to define the natural norms on the tensor products. In particular we have
De Rham
Complex
11
• For u in ft°(2Rm) ||w||2 = /
u{X)2 dX ,
(2.1.11)
where dX corresponds to the usual Lebesgue measure. By completion, we shall get the usual standard L 2 (JR m ) space. • For w in Q}{Mm) m
IMI2 = E /
.
"iiXfdX.
(2.1.12)
• For a in Q 2 (iR m ), we get IHI 2 = E /
^{XfdX.
(2.1.13)
These norms are of course associated to scalar products denoted by (• | •). We get of course {UJ | w) = | M | 2 .
(2.1.14)
Once we have this prehilbertian structure, it is immediate to associate to d a formal adjoint d*. More specifically we associate to each d^ defined from flp into flp+1 a formal adjoint d*'^ which maps f2 p+1 into QP and is uniquely defined, for w £ ilp+1, by the relation (d*iP)u | a/) = (u | d & V ) ,
(2.1.15)
for all LJ' e fig. This defines first d* " w as a distribution but explicit computations (integration by parts!) shows actually that d* is a differential operator of order 1 mapping explicitely fip+1 into fip. We get in particular the following formulas : • For a one-form CJ, we have
d*muj = -J2dXjcjj
,
(2.1.16)
12
Witten Laplacians
approach
• For a 2-form a, we have
(d*ma)k = -J2dXjajk,
(2.1.17)
3
where x \-¥ <Jjk(x) has been extended as a function with values in the space of the antisymmetric matrices. It is also easy to verify that d* is also a complex. This is shortly written as d*od* =0
(2.1.18)
which means that is, for p = 0, • • •, m, d*iP)od*iP+1)=0.
(2.1.19)
We are now able to define the corresponding Laplacians A ^ defined in a contracted way as A^
= (d + d*)2
(2.1.20)
and more explicitely, for any p, whose by A(p) = d*M o d<*> + d^V
o
(2.1.21)
The Laplacian A(p) maps fip(iRm) into itself. This is an elliptic operator of order 2 with diagonal principal symbol. All these constructions are valid when Mm is replaced by a Riemannian manifold. The operator A^°) is usually called the Laplace-Beltrami operator. In the case of ]Rm, the situation is particularly simple because we have the identity A ( P ) = A (o)
g, Id _
(2.1.22)
When p = 1, this gives
A « w = 5^(A<°>w,-) dxj .
(2.1.23)
j
Note also that m
A(°) = - A = - ^ ^ . 3=1
.
(2.1.24)
Witten
Complex
13
There is a natural correspondence between 1-forms and vector fields. In this correspondence d^ becomes grad or V and d* becomes —div. In the case of a compact riemannian manifold M the introduction of these De Rham Complexes and of the corresponding Laplacians leads to the so-called Hodge theory. A central role is played by these Laplacians for which a Predholm theory can be developed. In particular, they can be defined as unbounded operators on Z,2-forms. These operators will be selfadjoint, with compact resolvent. In particular, the spectrum is a sequence of eigenvalues with finite multiplicity tending to +00 with a corresponding sequence of eigenvectors forming an orthonormal basis.
2.2
W i t t e n Complex
The distorted exterior differential is d$ = exp - $ / 2 d exp $ / 2 .
(2.2.1)
The idea of Witten was to consider instead of the De Rham Complex the associated new complex. Note that d^u^d^u+^ud^^,
(2.2.2)
and more generally we have d$ = d + -d$ A .
(2.2.3)
It is clear that we keep the property of the De Rham Complex d® o d$ = 0 .
(2.2.4)
We keep the prehilbertian structure given by the L2 norm and can then define d j by duality (d%uj I u)') = (u) I < W ) .
(2.2.5)
For a 1-form, we get in particular d^cj
= d*Wuj + ^w(V$) .
(2.2.6)
14
Witten Laplacians
2.3
approach
W i t t e n Laplacians
We can define now the Witten Laplacians on the forms by Afc> = (d* + dl)2 .
(2.3.1)
All these constructions were initially introduced by E. Witten on a compact manifold and $ was a Morse function (that is with non degenerate critical points). His idea (that we shall perhaps develop later) was to relate some invariants of the manifold M with some indices of the Morse function $ at the critical points. These relations are called Morse inequalities. In the case of JRm, we get more explicitly Ag° := A<°> + \ | V $ | 2 - \A$
,
(2.3.2)
and A £ > := A^0) ® Id + Hess $ .
(2.3.3)
Note that we have the following important relation 4°>A|°> = A « 4 ° ) ,
(2.3.4)
which is only the explicitation on the 0-forms of d$ o (d$ + d\)2 = cf$ o (d
(2.3.5)
What is changed ? At least if | V $ ( X ) | —¥ +oo and 0 .
(2.3.6)
Semi-classical
considerations
15
But this property is rather clear by definition of A and integration by part. We observe indeed (at least for the compactly supported forms) the identity (A*w|w) = | l ( d * + d * M | 2 .
2.4
(2.3.7)
Semi-classical considerations
If we replace $ by &/h, we get when considering /i 2 A* a new family of /i-dependent Laplacians which were actually the Laplacians considered by E. Witten in the case of the compact manifolds. The parameter h plays here the role of the Planck constant in semi-classical analysis. By looking at this new Laplacian on the 0-forms, we get in particular Agi = -h?A + \\V$(X)\2 - ^AZ(X) •
(2-4.1)
This has the form of a Schrodinger operator -h2A
+ Vh
with Vh := V0 + hVi and V0(X) := j | V $ ( X ) | 2 .
(2.4.2)
The main idea of E. Witten was that the dimension of the kernels of these various Laplacians (immediately related by the Hodge theory to the Betti numbers) can be estimated from above by a rather crude semiclassical analysis of the lowest eigenvalues of these operators as h —> 0. If we observe that the minimum of Vo is obtained at the critical values of $ and that the minima are non degenerate if and only if the critical points are non degenerate the harmonic approximation will finally give a rather satisfactory answer. The only zeros eigenvalues of the approximating harmonic operators correspond to the minima of $ . We shall at least in this book explain how to compute and justify this harmonic approximation and refer to [HeSj2] or the books [He2] or [CFKS] for further information.
Witten Laplacians
16
2.5
approach
A n alternative point of view : Dirichlet forms
We now discuss another point of view which is unitary equivalent to the previous one but appears more natural in the context of Laplace integrals. The idea is simply that we start from the usual De Rham Complex but take for the definition of the adjoint the prehilbertian structure given by the scalar product associated to the norm in L 2 (2R m ;exp— $ dX). This gives a natural adjoint complex cf ,$ and one can similarly introduce associated Laplacians which are denoted by A^,'. As in Subsection 2.4, we obtain for example the operators A(°} = - A + V $ • V ,
(2.5.1)
and A<£] = A^
®Id+
Hess $ .
(2.5.2)
These operators have to be considered as unbounded operators on L 2 ( i R m , J R ; e x p - $ dX) and L 2 ( 2 R m , i R m ; e x p - $ dX). They also appear quite naturally when considering the quadratic forms u >->• q(i\u,u)
= J |Vu| 2 e x p - $ dX ,
(2.5.3)
which is called the Dirichlet form, and u^q£\uj,uj)=
f\dWuj\2exp-$dX+
f\d*^0)^uj\2
exp-fcdX.
(2.5.4) We have indeed the following relations (on the compactly supported forms) q$\u,u)
= (A^u
| w) L 2 (E m ; ex P -* dX) ,
(2.5.5)
and similarly 9«, (w,w) = {A%'w | w) L 2 (ffi m iil m. exp _ <E>dx) .
(2.5.6)
In this point of view, the techniques based on the Lax-Milgram Lemma and the Friedrichs extension are available for defining these Laplacians associated to the Dirichlet forms.
A nice formula for the
17
covariance
Remark 2.5.1 In the case of a compact riemannian manifold M, the identity* (2.5.2) or more precisely the corresponding "form" version written in the form q£\u,w)
= J2 l|0*i<"fc||2 + / (
Hess
$ < " ) • « e x p - $ dX ,
(2.5.7)
is known from the geometers as the Bochner-Lichnerowicz-Weitzenbock formula. It is actually usually written only for exact 1-forms. We said in the title that this point of view is unitary equivalent to the Witten Laplacian point of view. We simply observe the following relations Afc} = e x p - o Ag
o exp-- .
(2.5.8)
We note of course that u i-» exp | u i s a unitary map from L2(Mm) onto L2(lRm;exp-$dX). Having this unitary equivalence, one can transfer all the identities we have obtained for the Witten Laplacians and we get in particular the identity d<°>40) = 4a)d<0> .
(2.5.9)
When trying to use standard theorems (about Schrodinger or about compact injections) it is usually more easy to come back to the point of view of the Witten Laplacians which are defined on L2(Mm).
2.6
A nice formula for the covariance
We assume in this section that the condition fexp-$(X)dX
= l,
(2.6.1)
is satisfied. This can always be obtained by replacing the old
. exp _ $
dx)
,
(2.6.2)
tOf course, the formula in this case has to be modified suitably. In particular the Lebesgue measure is replaced by the Riemannian measure on M and Hess $ is replaced by Hess $ + Ric(M) where Ric(M) is the Ricci curvature on the manifold.
18
Witten Laplacians
approach
for a suitable 1-form w, we get, observing that dg = d(g — (g)), Cov (/, g) = J{g - *o;) exp - * dX ,
(2.6.3)
which leads to the question of solving / - ( / ) = d*<*u .
(2.6.4)
For the reader who is more familiar with the formalism with vector fields, one can rewrite these statements in the following form. If we have in mind to write an expression of the covariance in the form c
° v (f,g) = {Vg | v)L2(RmtRm.exp_^
dX)
,
(2.6.5)
for a suitable vector field v, we get, observing that Vg = V(g — (
(2.6.6)
which leads to the question of solving / - ( / ) = -div v + V $ • v .
(2.6.7)
Coming back to the language of the differential forms, we can try to solve (2.6.4) with an u> in the form u = du, and this leads to the following equation for u
US-*'}-
<™>
We have not at this stage presented all the tools needed for analyzing completely this equation (see Chapter 6) but let us present (sometimes too vaguely) the situation. As previously mentioned, ^4^' is an unbounded operator (whose domain D(A$' is contained in L2(Mm; exp —$ dX)) which is a selfadjoint extension (which can be shown to be unique) of the differential operator denned on Co°(lRm). This operator is positive (this is a Laplacian) and A$ + Id is an invertible operator which maps D(A^]) onto L2(Mm; exp —$ dX) and its inverse is a selfadjoint compact operator whose spectrum is a sequence of eigenvalues fin tending to 0 as n —• -t-oo, with a corresponding sequence of eigenvectors un which can be chosen as an orthonormal basis of % := L2(Mm;exp— $ dX). These eigenvectors are easily seen in the domain of A$' and under natural conditions on $ we can
A nice formula for the
covariance
19
also get that they are in S(Mm). One says more briefly that A^' is with compact resolvent. It can then be shown that A$ can be diagonalized in the same basis with the property that the eigenvalues An = X%!' (which satisfy A„ = — 1 + - M tend to oo. A$ being a positive operator, we know that \\' diately seen that A^
1 = 0.
> 0. But it is imme(2.6.9)
So J 1-4 ui (X) = 1 is a possible first eigenvector (or ground state) of A$' attached to X[' = 0. From general results on the Schrodinger operator, it is known (see ReedSimon) that the first eigenvalue (or lowest eigenvalue, or groundstate energy for mentioning other terminologies) is simple, that is that the dimension of the subspace Ker(A),> - X[') is one. We recall that ^ 4 ^ is conjugate to a Schrodinger operator. But this can be seen more directly in our particular case by the following argument. Let x in the kernel of A%' : A§'\ = 0. Taking the scalar product of this identity with x, one can justifies the integration by parts in order to get 0 = (A<£)X\x)H
= \\dx\\2.
(2.6.10)
We immediately get d\ = 0 and consequently x is constant. This shows that the dimension of the kernel of A$' is one. Now, in order to solve the equation (2.6.4), it is clear that the necessary and sufficient condition for solving A$ 'u = / is the orthogonality of / with the first eigenvector which is expressed by (/) = 0. But f = f— (f) satisfies this condition, and this explains why we can solve (2.6.4). For / in C 1 , such that df is bounded, we have seen that there exists u such (2.6.4) is satisfied. We get by differentiation and using the relation (2.5.9) with u> := du, df = A^u
.
(2.6.11)
There is for A$' an analogous Fredholm theory as for A$'. If we denote by X(n] the increasing sequence of the eigenvalues, we know that 0 < X{1] < X{21] <
(2.6.12)
20
Witten Laplacians
approach
In the case when A^ ' is strictly positive then A$ we can consequently write that
becomes invertible and
w = (41))~14f ,
(2-6.13)
and this leads to the "nice" formula for the covariance of two functions : Gov{f,g)
= ( ( 4 1 ) ) " 1 ^ / I dg)nm
,
(2.6.14)
where w-1' is the Hilbert space of the 1-forms with coefficients in the space L2(Mm; exp - $ dX) denoted before by L2(IRm, 2Rm; exp - $ dX). In particular we get for the variance : var (/) = (J
((A^r'df)
• df exp - $ dx\
,
(2.6.15)
and the Poincare inequality var(/)<(Ai1))-1||4r|||£(1).
(2.6.16)
This inequality is not optimal in general. Exercise 2.6.1 Prove the inequality var(/)<(A(°))-1|M/lli(I, •
(2-6.17)
Hint Observe that H4fl&(i> = «
,(0) ;
( / - (/)) I ( / " « •
(2-6.18)
Exercise 2.6.2 Show (at least formally) and using (2.5.9) that \P
< X{°} .
(2.6.19)
Let us now consider situations when an easy lower bound of A^ available. In the case when <5 is uniformly strictly convex, Hess $(X) > a > 0 ,
is
(2.6.20)
Notes
21
we observe the following inequality between selfadjoint operators* A{V = A^
®Id+
Hess $ > Hess $ > a > 0 ,
(2.6.21)
which is an immediate consequence of (2.6.20), the positivity of A$ , (2.5.2), and, using abstract analysis (extending the result mentioned in Ruelle (1969)), we obtain (A^)-1
< ( Hess S ) - 1 .
(2.6.22)
Exercise 2.6.3 Prove this inequality in the case of strictly positive symmetric matrices. The Brascamp-Lieb inequality is then an immediate consequence of (2.6.14) and says var(/) < (( Hess $ ) _ 1 d / | df)nm
.
(2.6.23)
Of course, together with (2.6.21), the inequality (2.6.23) implies the Poincare inequality. We shall see later the more difficult result due to Bakry-Emery [BaEm] that the condition of strict convexity (2.6.20) is sufficient for obtaining the log-Sobolev inequality.
2.7
Notes • The Witten Laplacians were introduced in a different context by E. Witten [Wit] as an elegant way to prove Morse inequalities on a compact manifold. • The appearance of these Laplacians in the analysis of Poincare inequalities is rather old, at least in Riemannian geometry. In this context, the Witten Laplacians appear as a Bochner Laplacian.
'•We shall say that two possibly unbounded selfadjoint operators satisfy the inequality B < A if D(A) C D(B) and if (Bu | u) < (Au | u) , Vu € D(A) . In our case one can show that it is sufficient to verify this property in the suitable dense set of the compactly supported 1-forms in J7 1 (l?" 1 ).
22
Witten Laplacians
approach
• The understanding of its role in the analysis of decay estimates is more recent. Although it is already implicitly present in [HeSj5], this role was explicitly emphasized by Sjostrand in [Sj8]. • Brascamp-Lieb inequality ([BraLi]) appears as an easy consequence of this point of view. This was first observed in [He6] (See also [NaSp], [Jo] for variations).
Chapter 3
Problems in statistical mechanics with discrete spins
We discuss in this chapter various classical problems in statistical mechanics. This shows in particular why it is natural to analyze Laplace integrals and spectral theory, also in the context of discrete spins. 3.1
The Curie-Weiss model
The Curie-Weiss model is one of the simple models in statistical mechanics which plays an important role in the mean field theory. We consider a iV-spins lattice a = (01,02, • • •, aN) with <jj € {—1, +1}. In order to take into account of the magnetic field, we associate to any a the energy -BB(
Y, °i-°i-B\ l
£ °i) • \l
(3-1-1)
Associated to this energy, we can define a measure HN,B on {—1, + 1 } ^ which will be the main object in consideration. The measure of {a} is then given by HN,B{{°}) = 2-Nexp-(^EB(a^j
,
(3.1.2)
where T is the temperature and k is the Boltzmann constant. We observe that when J = B = 0 we recover the iV-tensorial product of ^(5i + 5-i) : (5(^1 + <5_i)) measure on {—1, + 1 } ^ .
, which is the equidistributed probability
23
24
Problems in statistical mechanics with discrete spins
The corresponding partition function ZN, which is by definition the total measure HN,B({—1, +l} A r ) of {—1, + 1 } ^ , is then given by ZS = 2~N E CTe {- 1 ,+i}« ex P
-EB{
• (exp— u)t e { - i , + i } N « p ( K E i l i ^ ) 2 +(*r)- 1 5(Ei<«< J v^))" (3.1.3) with v = J/kT. A simple computation based on the easy identity exp a 2 = - L = / exp ( - 1 - + V2<)d£
(3.1.4)
gives the formula ZN(B)
= (exp-zv) - ^ — • y exp - / v | - f c o s h f a ^ + ^ ) J drj . (3.1.5)
We rewrite this integral in the form :
[W
r
V 27T
JR
ZN{B) = exp -v • \ — • / exp
-N$v,B(ri)dr)
(3.1.6)
with $I>,B(V) = - y - In cosh
(3.1.7)
As in Section 1.3, we are first interested in the so called thermodynamic limit -
lim AT _1 lnZjv JV-KX)
and according to formula (3.1.6), this is easily analyzed as a Laplace integral. In particular, an easy computation gives : -
lim N^ITLZN N-Kx
=inf$„n(»7) • n
(3.1.8)
A complete expansion of TV 1 In ZN in powers of jf as N —¥ oo can also be obtained by standard methods. We will recall these results in Section
The 1-d Ising model
25
4.2. In particular we consider the minima of the phase $^,s(r/) and observe that the critical points are given by the equation 77 = \2v • tanh
1—
B
(3.1.9)
In the case B = 0, the value u — | of the parameter appears critical. We observe indeed : (1) if v > | , the phase has two symmetric minima (double well situation) which are non degenerate, (2) if v = i , the phase has a unique degenerate minimum, (3) if v < | , the phase is strictly convex (single well situation) and has a unique nondegenerate minimum. The transition of the convex case to the non convex case will be analyzed on a simpler model in Section 4.2.
3.2
The 1-d Ising model
We consider now the following model HN(a) = -J
^2
or- or+i -B
l
^2
or ,
(3.2.1)
l
where we identify o\ and <7i+jy. Similarly to the preceding section, we introduce the measure JJ,N,B on {—1, + 1 } W by £JV,B(M)
== exp -/3HN(o)
,
(3.2.2)
with /3 = -^T. If one introduces K(r, T') = exp /? f JT • r' + | (r + r ' ) ) ,
(3.2.3)
for r, T' = ± 1 , we see immediately that we can write : exp -/3HN(o)
= K(o1,o2)
• K(o2,cr3) • • • K(oN-!,oN)
-K(oN,oi) . (3.2.4)
26
Problems in statistical mechanics with discrete
spins
We observe now that K(T, T') with T, T' = ± 1 can be considered as a 2 x 2matrix / e x p /3(J + B) exp - / 3 J \ V exp -/3 J exp 0{J - B))
^'
& j
which is usually called transfer matrix and will appear very useful in order to express statistical quantities. Let us first consider the corresponding partition function ZN =def
^2
exp -0HN(a)
.
(3.2.6)
ae{-i,+i}"
An immediate computation gives
J2 KN(
ZN=
(3.2.7)
ZN = Tr KN = Af + A^
(3.2.8)
where Ai and A2 are the two eigenvalues of K Ai = exp (3J cosh(!3B) + y / sinh 2 (/35)exp 2/3J + exp -2/3J) A2 = exp /3Jcosh(/35) - ^/sinh 2 (/?.B)exp 2/3J + exp -2/3J)
(3.2.9)
The fact that these two eigenvalues are not equal when (3^0 will be quite important. In more general models we shall see that this property will be a consequence of the Perron-Probenius Theorem or more generally of the Krein-Rutman Theorem. We then compute the free energy FN := - fcTln ZN = -fcTln (Af + A^) = - iVfcTln (Ai) - kTln (1 + A^/Af) .
(3.2.10)
At the thermodynamic limit when N —»• 00, observing that Ai > A2 we get / = lim FN/N
= -fcTlnAi .
(3.2.11)
iV—yoo
We observe that the speed of convergence to the thermodynamic limit is immediately related to the comparison of A2/A1 with 1. More precisely, the strict inequality A2/A1 < 1 permits to get exponentially rapid convergence in (3.2.11). This will explain why we shall later be interested in estimates
27
The 1-d Ising model
on the splitting in more general situations. We shall also compute in this very simple situation consider the correlation pair (as • ar+s)N
=def Z^1
a
^2
*'
CT
H-rexp -j3HN{a)
.
(3.2.12)
An easy computation gives again : (
= Zj1
Yl
rKr(T,r')r'KN-r(T',T).
(3.2.13)
T,T'=±1
We can rewrite (3.2.13) in the form (as • as+r)N
= Z^1 Tr (a 3 o K r o a 3 o KN~T) ,
(3.2.14)
where « 3 ( T , T ' ) =TSTIT>
,
is the so called third Pauli matrix. If we now denote by |1) and |2) the eigenvectors corresponding to Ai and A2, we finally obtain Jim (a, • as+r)N
= | ( l | a 3 | l ) | 2 + (A 2 /Ai) r |
(3.2.15)
iv—yoo
We observe that if B = 0, then | ( l | a 3 | l ) | = 0 and |(l|a 3 |2)| ^ 0. In this case the correlation function decays exponentially with r. More precisely, there exists a constant c > 0 s.t., as r —¥ +00, (as • o-s+r)oo ~ cexp -r/£
(3.2.16)
£ = (ln^/As))-1
(3.2.17)
with
which is called the correlation length. In any case we observe that ((<7S - (a^oo) • (as+r - (<7s+r)oo))oo ~ c exp -r/£ . Again we observe that the splitting between the two eigenvalues A2 and Ai of the so-called transfer matrix K plays a very important role in the asymptotic behavior of the correlations.
28
3.3
Problems in statistical mechanics with discrete
spins
The 2-d Ising model
We now present the two-dimensional Ising model. We consider a twodimensional n x m lattice of m rows and n columns, with spins ap = ± 1 occupying the vertices of the lattice. The interaction energy in a given configuration of spins is now given by Hn,m{a)
C3-3-1)
= - J J2 °P • °Q > P~Q
where Y1IP~Q denotes a sum over nearest-neighbor lattice points P and Q, and where we have assumed that the coupling J between nearest neighbors is the same for spins in rows and columns. We assume also that J > 0, i.e. that the interaction is ferromagnetic, so that the ground state has all spins equal to + 1 or — 1. The partition function is now : Z„,m =
^2
exp -—Hn,m{a)
= ^ e x p v^Qp-crQ
(ff)
,
P~Q
(3.3.2) with v = J/kT .
(3.3.3)
The free energy per spin ip in the thermodynamic limit is defined* by lim ( m n ) - 1 InZ n , m .
-tp/kT=
(3.3.4)
Tn,n—^oo
In order to evaluate Z„, m , Kramers and Wannier relate this quantity to a spectral problem for the 2 m x 2 m transfer matrix K defined by
K(a,a')
= exp f — ^ Z ^ ^ '
exp
( v^2,crk
' a'k )
fc=i * A precise definition of this limit is surely needed but we do not enter in the details (See for example the book by D. Ruelle [Rul]). Here we shall choose an order by taking successively the limit n —> oo and then the limit m —• oo. See also Subsection 4.10 for a discussion around this problem.
29
Notes
It is then a simple matter to get the following expression for the partition function : Zn,m=
Tr K n = 5 ^ / i i ( m ) n ,
(3.3.6)
where /ii(m) > faim) > ^z(m) > • • • > fi2m(m) are the eigenvalues of the matrix K. If we first take the limit as n —> oo, we have immediately lim n _ 1 l n Z n
m
= ln/ii(m)
(3.3.7)
and hence from (3.3.4) -4>/kT=
lim m _ 1 ln/xi(m) .
(3.3.8)
m—>oo
The explicit computation of the largest eigenvalue /zi (m) was done by Onsager in 1944. This explicit computation permits to show that as m —> oo the lowest eigenvalue becomes asymptotically degenerate for T smaller than some critical temperature Tc. More precisely, for T > Tc, the spectrum of K is simple and the spacing between two consecutive n **•* ^m' is of order 1/TO, in particular ln/xi(m) - ln/i 2 (m) > 1/C
(3.3.9)
and below Tc there are two such spectra with spacings of — In /J,J of order ^ but shifted with respect to each other by an amount of order exp —cm for some c > 0. In particular, we get the following splitting : ln/ii(m) — ln/Z2(^) < - e x p — cm ,
(3.3.10)
for some c > 0. The basic idea suggested by M. Kac and illustrated by heuristic computations was to relate these properties to tunneling properties observed in quantum mechanics. 3.4
Notes • The material presented here is quite standard and we remain close from the presentation given in [Hel2] where other examples are presented.
30
Problems in statistical mechanics with discrete
spins
• For explicit computations of models, we refer to Ellis [El], Parisi [Pa] and Thompson [Tho]. The book [Rul] is a nice presentation of Statistical Mechanics. We were strongly influenced when starting research in this domain by the course by M. Kac [Ka2]. • Note that for example through formulae like (3.1.5) questions relative to discrete spins may be transfered into other problems concerning continuous spins (See Kac [Ka2], Brunaud-Helffer [BruHe], Helffer [He7]). • The analysis of the transition at v = | for a Schrodinger operator with a potential like in (3.1.7) has been further analyzed in [HeMa].
Chapter 4
Laplace integrals and transfer operators
4.1
Introduction
This chapter is devoted to the study of statistical quantities associated with a family (indexed by the dimension m) of measures
exp ~&m\x)
dx^
on Mm, where $(m\x) is a phase (or potential) and h is as possibly small parameter. Sometimes this parameter plays the role of the temperature but this is not always the case.
4.2 4.2.1
Classical Laplace method Standard
results
Before to enter in more general questions it is perhaps better to recall very simple results concerning Laplace integrals. The first one is modelled on the stationary phase theorem : Theorem 4.2.1 Let $ be a real C°° phase defined in a neighborhood V of the closure of the ball B(0,1) in Mn such that • $ > 0 on 5(0,1) ; $ > 0 on dB(0,1), • $(0) = V$(0) = 0, • $ has a unique non degenerate minimum at 0. 31
32
Laplace integrals and transfer
operators
Let a be a C°° function defined in V and let us consider the Laplace integral I(a, $; h) = j
a(x) exp -$(x)/h
dx ,
JB(0,1)
where h e]0, ho]. Then, as h tends to 0, I(a, $; h) has the following asymptotic behavior
I(a,$;h)~h*YsaJ-hJ
>
t4-2-1)
with a0 = (27r)9 • a(0) (det Hess $ ( 0 ) ) " .
(4.2.2)
The proof is rather simple (and actually simpler than for an oscillatory integral). The assumptions permit to reduce modulo exponentially small contributions (in C(exp —eo/h) for some eo > 0) to an arbitrarily small neighborhood of 0. We can then use the Morse Lemma in order to write in new coordinates : $(a;) — J2v] —'• $(y)- We are then reduced to the study of I(b, $; h) which is easy by taking the Taylor expansion of b at 0. We are then reduced to the computation of very explicit integrals associated with gaussian measures on IRn. Lemma 4.2.2 (The Morse Lemma) Let f(x,y) (x £ Mn,y € JR ) be a real valued C°° function in a neighborhood of (0,0). Assume that (Vxf)(0,0) = 0 and that A = ( Hess xxf)(0,0) is non-singular. Then the equation f'x{x,y) = 0 determines in a neighborhood ofOa C°° function x(y) with x(0) = 0 and we have in a neighborhood
ofO f(x,y)
= f(x(y),y)
+
(Az\z)/2
where z = x — x(y) + OQx — x(y)\)(\x\ + \y\) is a C°° function of (x, y) at (0,0). The proof is standard. One of the basic problems in the context of statistical mechanics will be to control all this procedure with respect to the dimension. There is also an analog of the above theorem for a phase with more than one minimum and under suitable assumptions at oo, like the existence of the inequality *(a:)> 7 f N p - C »
VzelR",
33
Classical Laplace method
for some p > 0 and C > 0, one can also consider Laplace integrals over JRn. The assumption that $ is real can be made weaker if one assume holomorphy of a and $ and we then enter in the world of the saddle point method. 4.2.2
Transition well case
between
the convex
case and the
double
As already seen in the analysis of the Curie-Weiss model, it is interesting to discuss problems with parameter when the phase is deformed from a convex phase into a double well phase. The most simple case of such a situation is the phase denned on St x H-> <j>{x, 0) = (1 - 0)x2 + x4 ,
(4.2.3)
and, for a € CQ°(]R), one can consider the integral I(a,4>(-,9);h) over 1R and it is interesting to follow the behavior of this integral with respect to 8 e]0, oo[. For fixed 9, we can distinguish three cases : • for 9 = 1, I(9;h)=a(Q)hi (fMexp-x4dx + • for 6 < 1, 1(6; h) = 7T5 o (0) hi(l + 0(h)), • for 6 > 1,
I(9;h) = (2(0-1))-*
exp^h*
( a ( V ( * - l)+/2) + a(-y/(9-
0(h?))
x l)+/2) +
0(h)).
As 6 ~ 1, one has to be more precise in order to get a behavior in h uniform with respect to 6. This can be done by using as a reference the integral J(a) = J exp— (ax2 + x4) dx JR
where a is a new parameter a = h~2(6 - 1). We observe that • for a —»• oo, J (a) ~ 7T2 a" 2, • for a = 0, J(0) = JR exp -x4dx, • for a = —00, J(a) ~ ^/2n • exp ^- • a ~ 2 .
34
Laplace integrals and transfer
operators
It is then possible to prove that I(0;h) = ±hi-J{a)-(a{y/{0-l)+/2)
+ a{-y/{6 - l)+/2) + O(fci)) .
In a very simple situation this explains what is going on near the phase transition in the semi-classical limit. The role of the temperature in the phenomenon is played by the parameter 8. 4.3
T h e m e t h o d of transfer o p e r a t o r s
Our initial problem was to find a rather general approach for the estimate of the decay of correlations attached to "gaussian like" measures of the type exp-$(X)dX
(4.3.1)
on Mn with $ . One proof of this estimate (in the case when $ has a particular structure) is based on the analysis of the transfer matrix method, originally due to Kramers-Wannier, that we have already seen for the study of the Ising model. In this case, the transfer operator was just a matrix and this explains the terminology : "transfer matrix method". We come back to the brief description given in the introduction. We shall only consider the case when d = 1 and treat the periodic case, that is the case when {1, • • •, n} is a representation of Aper = 2Zjn2Z. We consider the particular potential $
*<»>(*) EE * ( * ) EE \ (jrV(Xj)
+ I'J-^+ilM
(4.3.2)
where we take the convention that xn+i = x\ and where h is possibly a semi-classical parameter which is sometimes chosen equal to one if we are not interested in the "semi-classical" aspects. More generally, we could more generally consider examples of the form : $
*(X) = I (T,(n*i) + H*i,*i+i)) I
(4-3-3)
where / is a symmetric interaction potential on IR x M. Let us mention however that the example (4.3.2) appears naturally in quantum field theory when the so called "lattice approximation" is introduced. For this special
Elementary
properties of operators with integral
kernels
35
class of potentials, we shall demonstrate that the informations given by the transfer operator method are complementary to the other approachs. We shall present in the next sections the "dictionary" between the properties of the measure h~% • exp— $h(-X") dX on lRn and the spectral properties of the transfer operator K y (which is also called Kac operator for some particular models introduced by M. Kac) whose integral kernel is real and given on M x M by TS f
N
,
i
V(x)
Kv(x,y)=h-iexp—^
1
\x-y\2
• exp- -^-
V(y)
•e x p - ^ .
(4.3.4)
By integral kernel (or distribution kernel), we mean* a distribution in X>'(2R2) such that the operator is denned from C§°(]R) into V'(IR) by the formula / (Kvu)(x)v(x) JR
dx=
Kv(x, y) u(x) v(y) dxdy , Vu , v€ C£°(1R) . JRxR
(4.3.5) This dictionary permits to obtain interesting connections between estimates for the quotient /^/Mi (immediately connected to the splitting by the relation : ^ — 1 = 777 (M2 — A*I)) of the two first largest eigenvalues of the transfer operator and corresponding estimates controlling the speed of convergence of thermodynamic quantities or the decay of the correlations. In particular, for fixed h, this speed of convergence can be exponentially rapid as n —> 00. For this we do not need convexity of the potential. The phenomenon is here similar to what we have observed for the one-dimensional Ising model. In particular there is apparently no phase transition. But all these problems reappear in the semiclassical context h —¥ 0.
4.4
Elementary properties of operators with integral kernels
We first recall some rather standard properties of operators with integral kernel. "Here we shall always consider much more regular kernels which are in particular continuous. So the notation J can be interpreted in the usual sense. In general, this means that the distribution kernel Ky is applied to the test function (x,y) i-> u(x)v(y).
36
Laplace integrals and transfer
operators
A natural criterion for verifying that the operator with integral kernel K can be extended as a bounded operator on L2(Mn; M) is given by the so-called Schur's Lemma : Lemma 4.4.1 Let K an operator whose corresponding integral kernel (x, y) \-> K(x, y) satisfies Mi := s u p x € E „ JRn \K{x, y)\dy < +00 , M2 := supyg^n JRn \K(x,y)\dx < +00 .
(4.4.1)
Then K can be extended from Co°(JRn) to a linear continuous operator on L2(M ) (still denoted by K whose norm satisfies ||K|| < yjMiM2 .
(4.4.2)
Proof By Cauchy-Schwarz inequality, we have |Ku(x)\ 2 < J \K(x,y)\\u{y)\2dy
J \K{x, y)\dy < M, J \K(x,y)\\u{y)\2dy
.
Integrating over x and using Fubini, gives then the result. Exercise 4.4.2 Write and prove an analogue for N x N matrices. When the kernel (x, y) H-> K(X, y) is also a symmetric kernel (we recall that we are considering only real kernels^) K(x,y)=K(y,x),
(4.4.3)
then the operator becomes a formally symmetric operator, in the sense that ( K u I v)L2 = (u I Kv)L2
,Vu,ve
C^(M) .
(4.4.4)
When K is in addition continuous on L2(M), we get a symmetric operator, that is satisfying (Ku I v)L2 = {u I K<;)L2 , Vu,v G L2(M) .
(4.4.5)
The next point is to ask under which condition the operator K is a compact operator on L2(M ; JR), that is the limit for the convergence in norm of t i n the case of complex kernels, one should consider the condition K(x,y)
=
K(y,x).
Elementary
properties of operators with integral
kernels
37
finite rank operators. An easy criterion is obtained by verifying that the operator is an Hilbert-Schmidt operator that is that the kernel satisfies* the condition \m\2H.s-=
\K(x,y)\2dxdy.
[
(4.4.6)
JRxR
We recall that Theorem 4.4.3 When the operator K is compact, symmetric and injective, then there exists a decreasing (in modulus) sequence /ij of eigenvalues tending to 0 and a corresponding sequence of eigenfunctions Uj which can be normalized in order to get an orthonormal basis of L2(IR). Moreover, the operator becomes the limit in norm of the family of operators K ' " ' whose corresponding kernel are defined by N
KW(X,
y)
= Yl WW
u
^y) •
(4-4-7)
We note indeed that
IIK-KWll^supl^l.
(4.4.8)
j>N
The symmetric operators 5 are called trace class if we have in addition the property that
\\K\\tr:=J2\H\ <+oo.
(4.4.9)
3
In this case, we get that
||K-KW|| t r < 5 3 | ^ | .
(4.4.10)
j>N J Note that more intrinsically, one can define the Hilbert-Schmidt operators as the bounded operators such that, for any orthonormal basis ei, the condition ^ \ | | K e i | | 2 < +oo is satisfied. It is actually enough to have the property for one orthonormal basis and we h a v e : | | K | | k s = £ . | | K e 4 | | 2 . §When K is not symmetric, trace class operators can still be defined by considering v K * K . Note that when K is trace class, one can compute the trace by considering any orthonormal basis e4 : TV K = y \ ( K e , , e j ) ,
38
Laplace integrals and transfer
operators
For a trace class symmetric operator, we can in particular define the trace as TV K = £ > , - ,
(4.4.11)
3
and this operation is continuous on the space of the trace class operators I IV K | < ||K|| t P .
(4.4.12)
This of course extends the usual notion of trace for matrices. We can actually compute directly the trace of a trace-class operator in the following way. We first observe that Tr K =
lim
Tr K ™
(4.4.13)
TV—>+oo
Then we observe that N
Tr K W = V V - = / K^N\x,x)dx J
j=i
.
(4.4.14)
R
We consequently obtain that TrK=
f K<~N\x,x)dx
lim N-++00
.
(4.4.15)
JR
If we observe that x H-> ^ . |/jj||uj(a:)| 2 is in ^(M), then it is natural^ to hope (but this is not trivial!) that x H-> K(X, X) is in L1 and that Tr K = / K(x,x)dx.
(4.4.16)
JR
Note that it is only when K is positive that the finiteness of the right hand side in (4.4.16) will imply the trace class property. ' L e t us sketch a proof of (4.4.16) in our particular case. K being positive and with an explicit kernel : exp ^ exp— J\x — y\2 exp | " , one can find an HilbertSchmidt operator L satisfying L*L = K. The kernel of L is given, for a suitable 0, by L(x,y) = eg exp— 9\x — j / | 2 e x p M^. We note indeed that c$exp—9\x — y\2 is the distribution kernel of exptgA for a suitable tg > 0. Then we observe that L is Hilbert-Schmidt and that | | L | | | j s = ^ . fij = Tr K . Using the previously mentioned formula for the Hilbert-Schmidt norm and the property that K(x,x) one obtains (4.4.16).
= I L*(x,z)L(z,x)dz
= J L*(z,x)
L{z,x)
dz ,
Elementary
properties of the transfer
operator
39
In the case the operator is Hilbert-Schmidt, that is with a kernel K in L 2 , then the operator is also compact and we have the identity \K(x,y)\2 dxdy .
5>i=/ •
(4.4.17)
JRxR
This gives an easy criterion for verifying the compactness of the operator.
4.5
Elementary properties of the transfer operator
We now describe the properties of the transfer operator (also called Kac operator in the special case considered by M. Kac) introduced in (4.3.4), by applying the criteria presented in the previous section. This operator is associated with a potential V. We do not try to give the minimal assumptions which are needed for V and we assume in order to start the discussion that V is continuous and satisfies : V(y)>±\y\2-C
(4.5.1)
for some strictly positive constant C. This implies in particular the HilbertSchmidt condition because 2
2 \Kv(x,y)\
/ ,a
dxdy < const. I / exp——V(x) dx 1 < +oo , (4.5.2) h
\J
J
and the trace class condition if we observe that, when I(x, y) = J\x — y\2, the operator is positive in the sense that ( K v u | u)L2 > 0 , Vu G L2{1R) .
(4.5.3)
This becomes the property that / exp -\x — y\2v(x) v(y) dxdy > 0 , JM2 4ft
(4.5.4)
with v = exp — ^u, and we get the result by observing that / exp-—-\x - y\2 v(x) v(y) dxdy = c(h) / e x p - / i £ 2 |w(£)|2 d£ (4.5.5) JlR? 4ft JR with c(ft) > 0.
40
Laplace integrals and transfer
operators
We then observe easily that the operator is trace class under the same condition on V. The operator K y is consequently a compact strictly positive selfadjoint operator.
4.6
Operators with strictly positive kernel and application
We observe now that the kernel (x,y) H-» KV{X, y) satisfies the condition Kv(x,y)
> 0 , Vx,y € IR .
(4.6.1)
In particular it satisfies the assumptions of the extended Perron-Probenius Theorem also called Krein-Rutman Theorem and our positive operator K y admits consequently a largest eigenvalue /ii equal to | | K v | | which is simple and corresponds to a unique strictly positive normalized eigenfunction which we denote by u\. Let fij the sequence of eigenvalues that we order as a decreasing sequence tending to 0 : 0 < fJ.j+1 < Uj < .... < fi2 < fJ-i •
(4.6.2)
We shall denote by Uj a corresponding orthonormal basis of eigenvectors with KvUj = HjUj ,
\\UJ\\
= 1.
(4.6.3)
Let us recall the proof and the statements : Definition 4.6.1 Let A be a bounded operator on a Hilbert space H — L2(X,dv) where (X, u) is a measured space. Then we say that A has a strictly positive kernel if, for each choice of a non negative function 6 £ H (||0|| ^ 0), 0 < AS ,
(4.6.4)
almost everywhere. Example 4.6.2
The two following elements are present in the book :
• in the finite dimensional case, the matrices with strictly positive coefficients, • on L2{JRm), the kernel ( 2 T T ) - ^ e x p - ^ e x p - i | a ; - y | 2 e x p - ^ .
Operators with strictly positive kernel and
application
41
The theorem generalizing the Perron-Probenius Theorem is then the following : Theorem 4.6.3 Krein-Rutman Theorem Let A be a bounded compact positive symmetric operator on % having a strictly positive kernel and let \\A\\ = A be the largest eigenvalue of A. Then A has multiplicity 1 and the corresponding eigenvector fi can be chosen to be a strictly positive function. Proof Since A maps real functions into real functions, we may assume that fi is real. We now prove that (i4n.fi) < (j4|fi|,|fi|). This is an immediate consequence of the strict positivity of the kernel. We just write : fi = fi+ - fi_ and |fi| = f i + + f i _ and the above inequality is then a consequence of (Afi+,fi_) > 0 , and (fi+,Afi_) > 0 . We then obtain A||fi|| 2 = (vifi.fi) < < | | 4 | ||fi|| 2 = A||fi|| 2 . This implies (^fi,fi> = (j4|fi|,|fi|) . This equality means (fi + ,i4fi_) + ( f i _ , A f i + ) = 0 .
42
Laplace integrals and transfer
operators
We then get a contradiction unless ft+ = 0 or ft_ = 0 . We can then assume ft > 0 and the assumption gives again 0 < (9, Aft) = \(6, ft) , for any positive 9. This gives ft > 0
a.e .
But ft = A^Aft and this gives, by (4.6.4), ft > 0
a.e .
Finally if two linearly independent eigenvectors \& and ft corresponding to A exist, we would obtain the same property for ^ by considering as new Hilbert space the orthogonal of ft in W. But it is impossible to have two orthogonal vectors which are strictly positive. Exercise 4.6.4 Consider the matrix case. Transpose the definition of the property of strict positive kernel and prove Theorem 4.6.3 in this situation. This is just the standard Perron-Frobenius Theorem. Exercise 4.6.5 Show that Theorem 4.6.3 is still valid without the assumption that A is positive. That is show that ||yl|| is actually an eigenvalue of A. 4.7
Thermodynamic limit
Let us first look at the thermodynamic limit. This means (cf (1.2.5)) that we are interested in the limit limn_).+oo n ^n ifxi™ e x P ~ ^K-^Q ^ 0 • ^ e s t a r t from the decomposition : e x p - $ ( X ) =Kv(x1,x2)
• Kv(.x2,x3)
• • • Kv(xn-i,xn)
•
Kv{xn,xi) (4.7.1)
and we observe that / exp-$(X)dX= jRn
[ K
(4.7.2)
43
Mean value
where Ky (x,y) is the distribution kernel of (Ky)". Our assumption on V permits to see that (Ky)" is trace classll and we rewrite (4.7.2) in the form : /
exp - $ ( X ) dX = Tr [(K v ) n ] = $ > ? ,
JR
(4-7.3)
j
"
where the fij are introduced in (4.6.2). We note also for future use that Kp\x,y)
= ][>>,(zK(y) .
(4.7.4)
j
In particular we get for the thermodynamic limit : In /•„„ exp -$(X) dX , lim — ^ — — = ln/xi . n—K»
(4.7.5)
n
Moreover the speed of convergence is easily estimated by : In f
- I n — sia2
exp-(X)dX
r
In Hi
(4.7.6)
- n l n ( ^ ) - lnfc2 + Inn + £>(exp ~52n) where k2 is the multiplicity of ^2- We observe here that the splitting between fix and \i2 determines the speed of convergence. 4.8
Mean value
Let us continue the dictionary. We consider now the mean value with respect to our measure of some "cylindrical functions". More precisely, if / is a function in CQ°(IR), let us introduce
JRn exp -$(X)dX An immediate computation using (4.7.4) gives
(/|11) ^
E^/( W r i 2^j Hj
II See for example [Rob2].
(482)
44
Laplace integrals and transfer
operators
In particular, we get the following convergence lim„= / f{y)ul{y)2dy,
(4.8.3)
and we have moreover the following control of the speed of convergence
{f{1))n
~Lf{y)ui{y)2dy=o{{ff)
4.9
(4 8 4)
•
--
Pair correlation
Let us consider the pair correlation functions. If / and g are two functions in Cfi°(]R), we denote by fW the corresponding function on Mn :
X -> f(Xj) . We shall now compute the covariance C o v ( " ) ( / W , 5 « ) = <(/(*> -
•
(If f(x) = g(x) = x, we write more simply Cor( n )(z,j).) Because everything is invariant by circular permutation we can take i = 1. We first have _ Sfc,tfcVrJ+1
(/(1)5W>n =
[/a/h/WdOMp)
T^F
In particular, if / = g (and this will be the interesting case for us), we get
,.(!).(,-). \T
J
)n
E M /4~ V r + 1 UR m My) My) dy? ^
n
i
and
(Eitf) 2 - Cor(")(/(!),/«) E fe ^(/ H /(y)«*(y) 2 di/) 2 ( E X ) - ;E fc ^a K /(2/)^(2/) 2 ^) 1+ 5+ (JRf(y)uk(y)ue(y)dyy + Efc^^ Vr
Q>?)
1-dimensional
lattices
45
and, taking the limit as n —>• oo, we finally obtain the following identity
lim Cov(")(/«,/«) = V
/ f(y)uk(y)ui(y)dy
J
JR
fc>l
. (4.9.1)
From this we get the inequality M2NJ
var(/)>lim
fll I
Cov^Uf^J^)
n->oo L 3-1
/
>
f{y)u2{y)u1{y)dy
(4.9.2)
2
(4.9.3)
JR
where var(/) = / ux(x)2 f(x)2 dx - ( f
Ul(x)
f(x) dx\
is the variance of / with respect to the measure on IR : u\{x) dx. For the first inequality, we have used the property that, by the Parseval equality, we have / (f(x)u1(x))2dx
= '^2 ( / f(x)ui(x)ue(x)dx)
,
remembering that Uj is an orthonormal basis. So we have shown that when d = 1 the correlation pairs tends to 0 exponentially rapidly as \i — j \ —>• +oo.
4.10
2-dimensional lattices
We consider now that A is a periodic sublattice A per in the form Z£/mZZ x Z/pZ and this means that the phase has the form $
W=
V X
Yl
J2 \Xh3~Xk,(
( ij) + J
(i,j)6AP
er
(4.10.1)
(*,i)~(fc,<)
Here we assume that v satisfies, for some constant C, \x-y\2
+ C).
(4.10.2)
46
Laplace integrals and transfer
operators
We can then try to follow the transfer matrix approach in one direction. This means that we consider that
*pr) = J2 v(m\xi) + ^ J2 \Xi - ^+ii 2 i=l
(4-10-3)
i=l
where the potential V = V( m ) is defined on Mm and given by m
rr rn
(with the convention that m + 1 = 1). We can try to take successively the limit p -»• +oo and then m —• +oo. It is then natural to compare what we get with the more standard limit procedure obtained by taking for example a square periodic lattice m = p and then taking the limit m —>• +oo. Let us look in this spirit at the thermodynamic limit. Following what we have done in Section 4.7, we start from the decomposition: exp-$(X)
= Kv(xi,x2)
• Kv{x2,x3)
•••Kv{xp-i,xp)
• Kv{xp,x{) , (4.10.5)
with V = V{-m\ and we observe that /
exp-$(X)dX= f
K$\y,y)dy
(4.10.6)
where Ky(x,y) is the distribution kernel of ( K v ) p . Our assumption on V (that is on v) permits to see that ( K v ) p is trace class. We can then rewrite (4.10.6) in the form : / m
exp -$(X) dX = Tr [(K v ) p ] = V n 3 { m ) P ,
J]R P
(4-10.7)
j
where the fij (m) are introduced are the eigenvalues of K y with V = V^m>. In particular we get : lim h / * - » " * - » ( * ) « * * p—»oo
p
= hft(ffl)
(4.10.8)
2-dimensional
47
lattices
and the speed of the convergence is easily estimated by : -In
InJ
exP-Z(X)dX
R
ln/ii(m)
p
, , .. = - p i n ( ^ g } j - Ink 2 (m) + Inp + 0(exp -S2p) ,
(4-10-9)
where k2(m) is the multiplicity of/x 2 (m). The problem is now that we would like to control the remainders with respect to m. We shall prove the following theorem : Theorem 4.10.1 Let c> 0 and let Q(c) C IN2 be defined by m 1 c < —<- . p c Let us also assume that the potential V has the structure (4-10.1), with v satisfying (4-10.2). Then we have the following identity lim
— In ( /
(m,p)-v+oo mp OO ,, (m,p)££l(c) (m,p)£S1(c) TUP
\JRmp \jRrnp
exp -$(X)dX\
)J
= lim
m-i+oo m-+ + a
ln(/xi(m)) m
(4.10.10) Let us first write
We now take the logarithm and divide by mp. We get
ME/ijM") mp
=
i"Mi(m) +
ln 1
( + E J > 2 (^g) P )
m
mp
The second term of the right hand side was already seen as tending to 0 as p -» +oo. But we now want to take first p = m and then make m —t +oo. From the direct analysis of the transfer operator, we can see (cf (4.10.9)) using (4.10.2) that under our assumptions the limit of ln ^ exists : lim m-»+oo
lnui(m) _ ^ U ; = £oo . m
(4.10.12
48
Laplace integrals and transfer
operators
We first verify by an easy comparison of V with a potential which is the sum of potentials with separate variables, that there exists two constants Si > 0 and 62 > 0 such that exp— Sim > Hi(m) > exp—52m .
(4.10.13)
£ 0 0 7* 0 .
(4.10.14)
We then easily get
We also observe the following identity concerning the trace of the transfer operator (if we assume h = 1 or J = 1). There exists /? > O such that, for all m, £ H {m)
=r
{J
exp-V^\y) dy^j .
(4.10.15)
Immediate estimates give the existence of a constant C such that, Y^Hiim)
< exp(Cm) , Vm € IN .
(4.10.16)
3
Here we do not try for the moment to estimate the constants C and 5 with respect to various parameters like J or h. We now come back to the estimate of the quantity —*
*
l
—L- for m = p, that is :
Ki+E^fe^n We observe that it is sufficient to find a constant C such that 0< Y
(ti^llY
<
e X p(C
m) .
(4.10.17)
But using the trivial estimate that /i J (m)//xi(m) < 1, we obtain that
and our estimate (4.10.17) is a consequence of (4.10.13) and (4.10.16). The general case considered in the theorem can be done in the same way.
Notes
4.11
49
Notes
Section 4.2 • The case when the space Mn is replaced by a more general measured space and the corresponding measure by a more general one is quite interesting : this permits in some case to treat simultaneously spin systems and classical integrals. Most of the techniques presented in this course are relevant only for the case of Mn (or for cases leading to this situation as it is the case in the next section). We refer to the book of R. Ellis [El] for a more general point of view. • This proof of Lemma 4.2.2 can be found for example in Hormander's book (Tome III p. 502) [Ho]. • We refer to the book of J. Dieudonne [Die] for an introduction of the saddle point method or to the monography by J. Sjostrand [Sjl] (Chapter 2) for a more sophisticated and general presentation. • The model (4.2.3) is analyzed in [He3] and corresponds to the analysis of phase transition. The phase which is analyzed there is actually (j>(x) = x2 — lncosh\//foSection 4.3 • The transfer matrix method was initially developed for the Ising model. In a more general context, this is presented in the books of Parisi [Pa] and [Tho] (see also [RoyC]). The dictionary presented here is developed aloung the lines of a course by M. Kac [Ka2] and more recently [He6] and [HelO]. Another interesting reference is the course by J. Frohlich [Fr]. • The "lattice approximation" is presented in [Fr] or in [GUa]. Section 4.4 • We refer to [RS-I] (Chapter VI) for this theory of operators with integral kernels. • Schur's Lemma can be found for example in [Ho] (Vol. Ill, p. 74). Section 4.6 • We have followed for the Krein-Rutman Theorem the presentation of Glimm-Jaffe (see Theorem 3.3.2 [GUa]).
50
Laplace integrals and transfer
operators
Section 4.7 • For the existence of the thermodynamic limit in the case of Laplace integrals, there exists an extensive literature (cf for example [LebPr], [Ru3], [BeHo] (Chapter VI) and [COPP]). Section 4.10 • The proof of the existence of the thermodynamic limit for transfer operators (through the verification of some subadditive property) is done in [He6]. See also [He9]. • This analysis was motivated by some paper by T. Bodineau [Bod2] on the Kac conjecture and is extracted from an unpublished paper by B. Helffer (July 97).
Chapter 5
Semi-classical analysis for the transfer operators
5.1
Introduction
We have seen that the analysis of limm_>.+00 ( ^ In f JRm exp — -—j^-dX J J with
^){x)
=
Y/v(xj) +
-j2\^-xj+1\2 3
.3 = 1
as h —> 0 can be reduced to the analysis of the transfer operator : V
V
Kv =exp-—exp/iAexp-^as /i —> +oo. This is what we call semi-classical analysis in reference to the analysis of the Schrodinger operator —h2A + V(x) where h corresponds to the Planck constant. In this semi-classical analysis, the harmonic oscillator plays an important role and it is not surprising that in the semi-classical analysis of the transfer operator we meet the corresponding "harmonic Kac operator". 5.2
Explicit computations for the harmonic Kac operator
In the context of the transfer operator it is interesting to analyze the case when V{x) = ax2/2 and we call this operator the harmonic Kac operator whose kernel is, for a > 0 and /? > 0 OCX
1
n
OLV
K(x,y):=exp-—-exp-—P\x-y\2-exp-^51
.
(5.2.1)
52
Semi-classical
analysis for the transfer
operators
A change of variable* make it unitary equivalent to an h independent operator (up to some multiplicative renormalization). We now suppose h = l. The analysis of the spectrum can be done in 4 steps • Step 1 We start from the kernel K{x, y) := exp —Q(x, y) where Q is a positive definite quadratic form, Q(x,y) = ^x2-bxy+^y2
,
(5.2.2)
with a = a + (3 and b = /3. We observe that a > b > 0. We first look for the ground state in the form u\{x) = (£)* exp—|a; 2 where t > 0 has to be determined. This is obtained by looking at the equation Kui = ni(t) ui .
(5.2.3)
The unknown t is obtained in the following way. If we consider the new quadratic form : (x, y) M- P(x, y) := Q(x, y) + -(y2 - x2) we have to express that dyP(x,y)=0.
(5.2.4)
This gives the following equation relating linearly x and y : y(x) = 7 ( i ) z , with 7 (t) = - ^
.
(5.2.5)
We have just to impose that P(x,y(x)) is constant (and then by computing for x = 0 identically 0), which is also expressed by the *If K is a transfer operator and if Uh is the unitary operator on L2(M) (Uhu)(x) = / i l t i ( / i 5 i ) , then the kernel K of Uh • K • U^1 is given by
K(x,y) = hiK(hix,hiy)
.
defined by
Explicit computations
for the harmonic
Kac operator
53
equation dxP(x,y)=0.
(5.2.6)
If (5.2.4) and (5.2.6) are satisfied, we have obtained that
P(X,y)=(^±iyy-y(X))l. This second equation (5.2.6) gives also the following relation between x and y : y{x) = 6
(5.2.7)
b
This determines the following value for t t = ^a? - b2 .
(5.2.8)
• Step 2 Once we have determined t, we immediately get the corresponding H(t), by writing / exp -P(x, y) dy = ^i(i) .
(5.2.9)
JR
We obtain
Mi(«) = f e x p - ^ ± V dy = (-^~)i .
(5.2.10)
* a-\-t The eigenvector being strictly positive, it is necessarily the ground state (according to the Perron-Probenius Theorem), that is the eigenvector corresponding to the largest eigenvalue of K. We have consequently found the "first" eigenvalue and the corresponding eigenvector. JR
• Step 3 We observe that u\(x) = {^)* exp—|a; 2 satisfies ui(a;)-Mxui(a;)=0.
(5.2.11)
Inspired by the case of the harmonic oscillator case (see Section 5.5), we then suspect that the eigenvectors are given by un = cn Ln~1ui where L = — -^+tx and cn is a normalization constant for obtaining
54
Semi-classical
analysis for the transfer
operators
\\un\\ = 1. This can effectively be done by showing (by a simple integration by parts) that, for v in S(M), we have LKv
= v~K.Lv ,
(5.2.12)
with v=
a+t b '
(5.2.13)
We consequently obtain V l Kui == v-nKLnUl
.
(5.2.14)
We obtain that Mn = i / - n + V i •
(5.2.15)
M2 _ o Mi a + Va2 - b2
(5.2.16)
In particular we have
• Step 4 Finally, we observe that (un)n€w is an orthogonal basis of L2{M) which is a standard result about the Hermite functions. We have finally proven the : Proposition 5.2.1 The spectrum of the operator K, whose kernel is given by (5.2.2) with a > b > 0, consists in 0 and the sequence of eigenvalues -n+l
-p/f^)
( _*y
(52I7)
Each eigenvalue is simple and the corresponding sequence of eigenvectors gives an orthonormal basis in L2(M). Exercise 5.2.2 2 .
2
Analyze the spectrum of the operator whose kernel is exp — %y .
Harmonic
5.3
approximation
for the transfer
operator
55
Harmonic approximation for the transfer operator
We come back to the semi-classical situation and would like to show how the computations given in the previous section may be useful for the harmonic approximation. The first thing to oberve is that according to our previous re1
2
mark, the spectrum of the operator of kernel Kh := h~iexp— exp — 2h
ex
P — ^2h
^-
ls
identical to the spectrum of the operator ax2 B\x — y\2 ay2 K i = exp — exp — ' - exp — — .
If (|:) T exp —t\ is the eigenvector associated to the highest eigenvalue y,\ of the operator K i , then ( ^ ) 3 exp— t^ is the eigenvector associated to the same eigenvalue Hi which is also the highest eigenvalue of the operator
Kh. Let us now consider the more general transfer operator Kh whose kernel ex is given by Kh(x,y) = h~? exp—2h exP ~^2h P — 2 h a n ^ ^e* u s assume that V(x) > 0, tends to +oo at oo and that the infimum is obtained at at least some point.
According to the properties of this transfer operator, we first observe that the following Maximum principle is true. Proposition 5.3.1 Hi =
sup veL*(R)
(Kv | t/) L 2 (JR) .
(5.3.1)
, \\v\\
Proof. We have just to write +oo
(5.3.2) #
Application One can first try to use this very easy principle for getting a lower bound for Hi. The strategy is to find a trial function Vh of norm 1 in L2(M) with the idea to get (Khi>/i | Vh)L2(li) maximal. This problem could appear difficult
56
Semi-classical
analysis for the transfer
operators
for fixed h but becomes finally easier as h —>• 0. Before to enter concretely in estimates, let us list some possibilities for treating the problem. The following ideas are natural for analyzing the situation as h —> 0 : • Look for a vh in the form VH = c(h) exp —^-\x — a(h)\2. In other words this corresponds to the idea that we have to optimize over gaussians. • Look at the harmonic approximation at a suitable point and use the results obtained in the quadratic case. • Try to find an approximate eigenvector in a form of a gaussian. • Try to find an approximate eigenvector in the form of a WKB solution, that is with the structure a(x, h) exp —(x; h)/h. Each of these ideas leads to some information which has an influence on the other approachs. So it is may be useful to analyze each of the possibilities. In the two last approachs, we can hope not only a lower bound for fii but a localization of /xj. We can also hope to get the existence of other eigenvalues. First approach One has to look at the integral = h-1 J exp - vW+vM+J\*-vr+*\*-*°\2+'\v-M'
dxdy
,
&•*•*>
and ask for which choice of XQ and t the integral 1(h) is maximal. If V admits a minimum at x = xmin the choice of XQ = x m ; n seems rather optimal. The minimum of the phase is then 0 and attained at the point \X^y)
=
yXjnin^Xjjxijij.
The next question is then to decide which point to choose in the case there are more than one point xmin realizing the minimum. At least, when they are isolated, this suggests to try the various gaussians attached to the various minima. Then we can apply the Laplace method which is always available when t > 0. We shall get a result of the form :
I(h;t) = a(t) + Ot(h).
(5.3.4)
Harmonic
approximation
for the transfer
57
operator
The next step is then to optimize over t and possibly over the various xmin. The choice of the optimal t is related to the optimization of the determinant of the Hessian of the complete phase (x,y)»$(x,y;t): = \ (V{x) + V(y) + J\x - y\2 + t\x - xmin\2
+ t\y - xmin\2)
K
' ' '
at \xmin, xmin j. Second approach In the first approach, we were rather vague on the choice oft, just suggesting a strategy. We keep from the first approach that the main contribution is localized in a neighborhood of x = xmin. This suggests to replace the potential V by its quadratic harmonic approximation at x = xmin. This seems at least a good idea when V"(xmin) > 0, that is when V has a non-degenerate minimum at x = xmin. Then it is quite natural to take the particular Gaussian which is the eigenvector computed for the Kac harmonic operator. We obtain in this way, for a suitable t computed in the quadratic case (5-3-6)
I ( M ) = /ii(t) + Ot(>»),
where /xi(i) is the largest eigenvalue of the Kac harmonic operator with potential V"(xmin)(x xmin)2/2. Third approach We start again from the previous eigenfunction
t \x - x„ x i-> Uiarm(x; h) := h " e x p - — 2h
|2
of the Kac harmonic approximating operator. Then we hope to show that the function x ^ ulpp := x((x - z m i „ ) / ^ K a r m ( z ; h) , where x = 1 in a neighborhood of 0 and with compact support and 6 > 0, is an approximate eigenvector in the sense that K < p p ( x ; h) = mWufix;
h)+rh,
(5.3.7)
58
Semi-classical
analysis for the transfer
operators
with ||rh||L*(«)=o(l),
(5-3.8)
as h —> 0. Moreover, one has to verify that, if S < | , then \\u°*p{x;h)\\ = l + o ( l ) .
(5.3.9)
This means that the cut off occuring in the definition of u° p p keeps most of the L 2 -localization of u\arm. Let us also observe that in the two first approachs we had only to control (r/i | «iPP)z,2(i?) instead of (5.3.8). The verification of (5.3.8) is left as an exercise but it could be easier for the reader to look at the case of the Schrodinger operator in Section 5.5 first. Let us recall the spectral Theorem. Theorem 5.3.2 If P is a selfadjoint operator on an Hilbert space 7{, then, if A ^ S ( P ) , where S(P) is the spectrum of P, then
ll^-*'"'^ 3(3^5-
(5 3 10
'' '
We apply this theorem in the very simple particular case of a compact, self-adjoint operator K and we can more explicitly write :
We apply (5.3.7). If ^i(i) is an eigenvalue of K, we are done. If now n\{t) is not an eigenvalue of K, we have
inffo - Mi(*)) < A L = o(l) . J
llui
(5.3.12)
II
Consequently, we know in any case that in an interval of size o(l) centered at Hi (t) there is at least one eigenvalue of K. A b o u t the link between eigenvectors and approximate eigenvectors As one can understand from the study of the double well, one has to be very careful with this point! The following computation gives sometimes a
WKB constructions
for the transfer
operator
59
useful information when one knows a priori that there is a simple eigenvalue which is "sufficiently" isolated in the spectrum. Let us start from
and let us consider a normalized eigenvector u\ attached to an eigenvalue A of P which is assumed to be simple. Let Tl\ the orthogonal projector on u\. An immediate computation gives
\\uapp - Kxuapp\\ < , ( v J ^
} UA})
•
(5-3.13)
When the r.h.s. in (5.3.13) is small, then this gives a good information between uapp and a possibly renormalized eigenvector. 5.4
W K B constructions for the transfer operator
This corresponds actually to the forth approach. One looks at approximate solutions of the type x(x) phase 4>{x;h) ^^^(x)^
ex
P—
h
where the
,
3
is more adapted to the operator K than the Gaussians. This candidate as approximate eigenvector is only relevant for the approximation of the ground state. This is indeed the unique strictly positive eigenvector. We look formally for a solution of K exp
4>(x\ h) /^ A K -j—t ~ I 2 ^ fii,ih*l exp
<j){x\ h) ^-^ .
(5.4.1)
By formally, we mean more precisely that we apply the Laplace method around the critical point y(x), which is defined near some minimum x m j n of the potential V, by Mv) + \v'(y) + J(y-x)=0.
(5.4.2)
Here the principal term of the phase o{x) has to satisfy 1 1 -V(x) + -V{y{x))
J + 4>0{y{x)) + — \x - y{x)\2 =
(5.4.3)
60
Semi-classical
analysis for the transfer
operators
The constant is immediately seen as equal to 0 by computing everything for x = 0. So we get \v(x)
+ \v(y{x))
+ <j>0{y{x)) + j\x
- y(x)\2 = <j>0{x) .
(5.4.4)
Differentiating (5.4.3), we get the equation -#>(*) + \V{x)
+ J(x - y{x)) = 0 .
(5.4.5)
Then we get the system (5.4.2), (5.4.5), which we call eikonal equations in reference to what is currently done in geometrical optics. The system takes the form K(x,'0(x)) = (y,'0(y)),
(5.4.6)
where K is a map denned in a neighborhood of (xmin, 0) whose graph is the set of the points (x, £, y, rj) such that
v + \y\y) + J(y-x) = o, -Z+\V'{x) + J{x-y) = Q. We note that the map K is well defined*, invertible and that (x m j„,0) is a fixed point of K. Moreover this fixed point is an hyperbolic point, that is its differential at (z m j„, 0) is a matrix which has two eigenvalues 71 and 72 with |7i| < 1 and I72I > 1. In this case, there is a natural set defined in a suitable neighborhood V C M2 of the fixed point A = {(z,0eV|
lim K,W(x,Z) = {xmin,0)}.
(5.4.7)
n—>+oo
It can be shown (by a general theorem on hyperbolic maps) that A is locally a manifold which can be parametrized by x. This means that there exists locally a function x H-> g(x) such that A = {(*,£), £ =
(5-4.8)
Moreover K being a symplectic map (which means in our particular case that the determinant of the derivative of K at any point is equal to 1), one can also show that A is a Lagrangian manifold (this is in any case true when n = 1 as here, but we mention the argument because it can be used tStart by solving the second equation.
WKB constructions
for the transfer
operator
61
when n > 1) and o(a;) is the phase such that o(^min) = 0, (f>o(x) > 0 and 'o(x) =9(x). Remark 5.4.1 The equation (5.4.4) takes the form M*)
= Mv(x))+Oo{x),
(5-4-9)
with 0o(xmin) = 0. This leads to the formula
M*) = J2 6o(y{n)(x)) ,
(5.4.10)
neJV
with y(n^ defined by yl°\x)
= x , yln\x)
= y(y (n - 1J (a;)) •
(5-4.11)
We note here that y(x) is the first coordinate of K(X, <j>'0(x)). So (5.4.9) represents a "discrete" integral along the "trajectory" y^n\x)n&if going from x to xmin which is the projection on IRX of the "trajectory" K,(n\x,(/>'0(x)) inside the Lagrangian A going from (x,<j/0(x)) to (a; m i n ,0). This does not help so much at this point because the knowledge of 4>o (x) and y(x) are obtained simultaneously. Nevertheless, if we know try to determine j (for j > 1), we shall get equations in the form
M*) = Mv(*)) + 9M>
(5-4-12)
with 8j depending on the knowledge of e (£ < j) and of the coefficients of the expansion of the approximate eigenvalue. As in the case of the Schrodinger operator, this is the condition 9j(xmin) = 0 which determines the coefficient fj,^ recursively (with respect to (.). According to the contractant character of x H-» y(x) in the neighborhood of xmin (one can verify that 0 < y'{xmin) < 1) the condition vj \Xjnin)
= U
permits to make convergent in C°° the series
^(*)=£>(y(n)0i0). ngJV
in a suficiently small neighborhood of
xmin.
(5.4.13)
62
Semi-classical
analysis for the transfer
operators
Remark 5.4.2 This is of course strongly connected with what was done in the case of the Schrodinger operator. In order to find a so-called WKB solution of —h?d2/dx2 + V(x) — A, one first get the so-called eikonal equation <j>'0{xf = V{x) ,
(5.4.14)
which is explicitely solvable in the case of dimension 1. The general equation obtained when trying a formal solution of the type exp — 4>{x; h)/h with <j>(x; h) ~ Ylj h:icf>j(x) and A ~ ]T XjhP, is -4>'(x; hf + V(x) - A + h4"{x\ h) ~ 0 .
(5.4.15)
So (5.4.14) is obtained by writing the cancellation of the coefficient of h° in the left hand side of (5.4.15). Let us write the next equation corresponding to the cancellation of the coefficient of h. We get -2'0 • [ +
- Ai .
(5.4.16)
One can solve locally this equation if and only if Ai = $(&„*„) .
(5.4.17)
In the general case (n > 1), the existence of a solution of the eikonal equation which takes the form
|V<Mz)| 2 =^(x)-A 0 , is a consequence of a general theorem called the stable manifold Theorem which is the analog for Hamiltonian flows of the previous theorem used for the hyperbolic maps. The main point is indeed to look at a Lagrangian manifold A contained in the hypersurface in Mn x Mn :
e - V(x) = A0 defined in a neighborhood of (x m j n ,0) as the set of the points (x,£) such that $t(a:,£) -> {xmin,Q) as t -» - c o .
The case of the Schrodinger
5.5
operator in dimension
1
63
The case of the Schrodinger operator in dimension 1
We start with the simplest one-well problem : S£ := -h2d2/dx2
+ v(x) ,
(5.5.1)
where v is a C°°- function tending to oo and admitting a unique minimum at 0 with v(0) = 0. Let us assume that (5.5.2)
v"(0) > 0 .
In this very simple case, the harmonic approximation is an elementary exercise. We first consider the harmonic oscillator attached to 0 : ~h2d2/dx2
1 + =-v"(0)x2 .
(5.5.3)
This means that we replace the potential v by its quadratic approximation at 0 ^v"(0)x2 and consider the associated Schrodinger operator. Using the dilation x = h^y, we observe that this operator is unitarily equivalent to h -d2/dy2
+
\v"(0)y2
(5.5.4)
Consequently, the eigenvalues are given by Xn(h) = h • Xn(l) = (2n - l)h •
v"(0)
(5.5.5)
and the corresponding eigenfunctions, called the Hermite functions (when v"(Q) = 2) are
U*(X) = h~u\ ( JJ
(5.5.6)
with*
*We normalize by assuming that the L 2 -norm is one. For the first eigenvalue, we have seen that, by assuming in addition that the function is strictly positive, we determine completely u^(x).
64
Semi-classical
analysis for the transfer
operators
Here the Hermite polynomials (see also after (5.2.11)) Pn(y) can be defined, up to a multiplicative constant by : Pn(y)
= e x p y (K A _ 2 y/ ) n - i e xy p - ^ dy ' 2
(5.5.8)
We are just looking for simplification at the first eigenvalue. We consider the function u1'app' x i-4- x(x)ui(x)
= c • x(x)h
*exP —
v"(0) x2 ~~2~2h '
where \ is compactly supported in a small neighborhood of 0 and equal to 1 in a smaller neighborhood of 0. We now get S*-h-
v"(0)
uhl'app- = 0{hi)
.
The coefficients corresponding to the commutation of S„ and x gi y e exponentially small terms and the main contribution is (v(x) -
l
-v"{Q)x2)x(x)uh1{x) L2
which is easily seen as 0(h 2). Then the spectral theorem gives the existence for S% of an eigenvalue X(h) such that X(h) - h •
v"(0)
.
In particular, we get the inequality
Xi{h)
.
(5.5.9)
Combining with other techniques (see for example what will be done in Section 5.6 in the case of the transfer operator), one can actually prove that Xi(h)-h-
v"(0)
(5.5.10)
Harmonic
5.6
approximation
for the transfer operator: upper bounds
65
Harmonic approximation for the transfer operator : upper bounds
We have already presented the quadratic case. The aim of this section is to prove in the same spirit why the harmonic approximation leads to an upperbound in the case of the transfer operator. The first point is a general formula associated to any partition of unity
3
We have + \ £ ; M f c ) -
K
' ' '
We now take an adapted partition of unity. Let 8 be a function in CQ°(] — 1, +1[) such that 0 < 8 < 1 with 8 = 1 on ] - 1/2, l / 2 [ . We now assume that there is a finite number of non-degenerate minima. To simplify the exposition, we consider the symmetric double well situation and suppose that we have two minima at x'c = —s and xc = +s. We introduce forj € { - 1 , 1 } ,
(5.6.2)
and we choose 8 and <po such that the following relation is satisfied
J2
^(t,h)2 = l.
je{-i,o,i}
Let us first control the error term. Starting from
/ Jo
V
(5.6.3)
66
Semi-classical
analysis for the transfer
operators
we get the estimate (
< \* ~ V? • I Jo V^-(te + (1 - t)y)dt\2 <\x-y\2-^\VVi{tx + {l-t)y)\Ht.
Summing up over j , we obtain ^ | V ^ ( t e + ( l - % ) | 2 dt. Jo
For our partition of unity, we get the existence of C such that, for all he]0,ho], Y,{
(5.6.4)
The remainder estimate is then obtained by giving an estimate from above in C{L2) of the operator whose distribution kernel is (x,y) M- Ify(x,y)
:= ^ ( ^ ( s ) -
.
j
This is analyzed by Schur's Lemma (Lemma 4.4.1) saying that for a symmetric kernel \\Kv\\ciL*)<sup(J\Rv(*,v)\dv We have consequently just to compute
•-••sH^-rf+^u. 2 J
/i~ 5 / \x — y|" exp Jv
which is estimated from above by |2
h I / \x-y\2exp-^[^-^
1 (V{x) , J. 2 + ^-\x-y\ )
dy .
J Wt
This gives Ry(x,y)dy
) < C7i» e x p -
2h
and finally \Rv\\c{L')
< C^ .
(5.6.5)
First conclusions about the splitting
67
Terms corresponding to the harmonic approximation We now take j = ± 1 and consider the operator K ^ restricted to functions 2
with support in the interval \x — xc\ < h*. We then observe using the harmonic approximation that (K^(p±1u,(p±1u)
<expC7ii (Kvotp±1\u\,
.
.
(
'
It remains now to estimate from above (Kyipou \
.
(5.6.7)
Putting everything together, we have proved the estimate (K$u | u) < /x?(l + 0(hi))|H|2
.
(5.6.8)
This estimate is quite analogous to what could be obtained for the Schrodinger operator.
5.7
First conclusions about the splitting
We have finally proved the following theorem : Theorem 5.7.1 Let V be a C°°-potential satisfying V > 0 and admitting a finite number of non-degenerate minima Xi (i € I) such that V(xi) = 0. Let us associate to each Xi the Kac harmonic operator (h = 1) K;""™ corresponding to the potential Vi(y) := ^V"(xi)y2 and let us denote by /J.^ the largest eigenvalue o / K j . Then we have lim Hi(h) = sup/x^ . h-yO
(5.7.1)
i
Without to much effort, we get also : Theorem 5.7.2 Under the same assumptions, we have two cases for the splitting as h —» 0. • If there exists a unique io such that (i)
(io)
s u p / 4 ; =Mi
68
Semi-classical
analysis for the transfer
operators
then lim n2(h) = max I sup/if ) , fi2io)
1 .
(5.7.2)
In particular
lim ^ S
> 0.
(5.7.3)
fc-s-o H2(h) • / / not, that is if there exists IQ and i\ with %Q ^ i\ such that (0
sup Mi
(*o) = A4
(»i)
=Mi
i/ien we /iawe lim ^ g h-X)
= 1.
(5.7.4)
H2(h)
Of course these results are very rough but a more precise answer in the second case may depend of a deeper analysis based on the analysis of the so called tunneling effect. One typical case of the second situation is the case of the symmetric double well. In this situation we shall improve strongly the result in Theorem 5.9.1. 5.8
Some elements about the decay
One of the first elements occuring in the analysis of the tunneling is the analysis of the decay of the eigenfunctions. In the case of the Schrodinger operator, this is done through the so-called Agmon estimates. There is something similar occuring in the case of the transfer operator. At least if we are just interested in very rough estimates, we can just look at the identity KhUi = fii(h)ui .
(5.8.1)
It is then rather easy to see that ui(x; h) is exponentially small outside the minima of V. We now write that u\ is the first normalized eigenfunction in the form ui(x;h) = —TTT
Kv(x,y;h)u1(y,h)dy
.
(5.8.2)
Some elements about the decay
69
Using Cauchy-Schwarz in the y variable and the existence of a strictly positive constant a, such that Hi(h)>a,\/he]o,h0},
(5.8.3)
we can rewrite (5.8.1) in the form
O < Ul{x]K) < a~l ( I Kv(x,y)2dyY
.
(5.8.4)
This gives immediately the following estimate 0
-exp-^^ .
(5.8.5)
This means that the eigenfunction is strongly localized near the minima of V as h —> 0. This will permit to use cutoff function in order to construct quasimodes attached to each well. We would like now to improve the decay in the semi-classical situation. It is reasonable for example to think that the optimal decay is measured by comparison with exp — ^ where (j>a is the WKB phase (at least near a point where V is minimum). Although it is explained here in dimension 1, the argument is actually not limited to this case. Our starting point is (5.8.5). We immediately observe that we can implement this upper bound in (5.8.2) and we get the upper bound 0
exp-4>^(x)/h
with
4>W(x) = ]-V(x)+mil(x-yf 2
v
+ V(y)
We see that it is strictly better, and we observe the inequalities \v{x)
< W(x) < lv[x)
+|
dist (x, {V-\0)})2
.
One has actually a natural construction in order to get this function. We define (f>(°> = -^ and <j>^ as before and introduce by recursion on n £ IN (n)
(X)
V(x)
inf y
'V(y) , J ,
y\' + <
(n-l)
(y)
70
Semi-classical
analysis for the transfer
operators
It is easy to see that this sequence is increasing with respect to n and that \v{x)
< <^\x)
< l-V{x) + £ dist (x, {V~\0)}f
.
We can then define the limit 4>°° (x) which satisfies 0°°(x) =
^ + i n f ^ 2
y
+ ^\x-y\2
+
^(y)
Note that near one minimum of V, this limit <j>°° is actually the phase o constructed as solution of the eiconal equation near the same minimum. 5.9 5.9.1
Splitting revisited Preliminary
discussion
We first observe that, in the strictly convex case, there exists only one minimum and the harmonic approximation is quite sufficient for detemine the main behavior for the splitting. We concentrate now our study on the double well case. There are different variants of the so-called Hering formula which permit to analyze the splitting. We shall get rather easily by this approach an upper bound of the splitting n2 — £«i • It will take more effort to get the lower bound which seems to us new. 5.9.2
Comparison
of various
problems
Once we have obtained the decay of the eigenfunctions for the operator K y , one can, in the double well case, use the first eigenfunction u\ of K y in order to construct a quasimode for a one well problem K v 0 with Vo coinciding with V in the neighborhood of the minimum xc (and with VQ > V). It is sufficient for that to analyze (K V o - Mi)(X"i) where x i s a cut off function with compact support in a neighborhhod of xc and equal to one in a smaller neighborhood of xc. Then \\xm\\ = -j=+0{exp-eo/h)
(5.9.1)
for some eo > 0 due to the exponential localization of u\ established through (5.6.8) and the property that u\ is symmetric (because v is symmetric).
Splitting
71
revisited
On the other hand KVo(x«i)=Mi(x«i)+»-i
(5.9.2)
where
n(x; h)=h-2 /exp - ^ [V(x) + V(y) + f \x - y\2](x(x) - x{y))u\{y)dy (exp -e0/h)
,
(5.9.3)
for some strictly positive eo which is arbitrarily close (but strictly smaller)
V(x)+2V(y) inf (i,v)e supp ((x,y)^x(x)-x(y))
+ ^\x-y\2
.
In particular, we get, using the spectral theorem 5.3.2 the existence of an eigenvalue of K y 0 which is exponentially close to n\. We note that these arguments are not limited to the first eigenfunction and we have only used that u was an eigenfunction attached to an eigenvalue /x(/i) satisfying fj.(h) > a > 0. In the case when we know that there is a gap in the spectrum (more precisely if /iX is the unique eigenvalue for K v 0 in a fixed open interval containing /xi), we can also obtain that a normalized corresponding eigenfunction UQ of K y 0 can be chosen such that ||u 0 - \/2(xu{)\\L2 5.9.3
= O(exp—eo/h) , for some e0 > 0 .
Upper bound of the
(5.9.4)
splitting
The classical idea is here that the first eigenfunction will give us a natural way to construct a good quasimode corresponding to the second eigenvalue. We assume that V has only two non degenerate minima at ±xc and we assume xc = 1. Let us introduce r _ / ui ~ \ 0
for a; > 0 for x < 0 .
We define also ue by u (x) = ur(—x) .
72
Semi-classical
analysis for the transfer
operators
Of course we have ui=u\
+ u[
and we now claim that ap
T
u^ = U-, —1 >u is a good quasimode corresponding to ^2 and orthogonal to u\. minimax Principle, we have indeed
By the
M > (Ku?\ua2p) ^ 2fflR+xiR+(K(x>y)
-K(x,-y))ui(x)ui(y)
dxdy ,
that we can compare with /xi = (Kui|ui) =
2
f
IR+XR+
(K(X> y) + K(x'
-y))ui
(x)
u
i (y)dx
d
y •
We immediatly obtain the inequality A*i - M2 < 4 / / J
if (a:, -j/)ui(a;) «i(y) dxdy .
(5.9.5)
JR+XR+
The term 4 / JR+xR+ K(x, -y)m(x)ui(y) dxdy is easily controlled using the different estimates already obtained. We have, using (5.6.8),
f JR+XR+ K(X' -y)Mx) My)
dxd
v
+ i(x
+ yr]
dxdy.
We then obtain easily the upper bound Hi-fi2
1
exp-[^l-
V
1
inf
V(x)+V(y) + ^(x + y)2
2h(x,y)£R+xR+
This gives an exponentially small splitting in Oa \h
S(J) =:
(5.9.6) x
exp —Sjjjp-) with
2 ± y(x)+y(2/) + ^ ( x + 2/)
inf (x,y)€R+xR+
2
Here we emphasize that the upper bound is not optimal because we do not use the improved estimates on the decay of Ui ouside the wells. It is however clear that S{J) > 0. If J is small, we get the asymptotic
S{J) = J + 0(J2)
Notes
73
because the minimum of M+ x IR+ 3 (x,y) M- \V(x) + V(y) + ^(x + y)2} is attained at a point which is quite close, as J —> 0, to (x = 1, y = 1). We have proved the theorem : Theorem 5.9.1 Let us assume that V is a symmetric double well C°° potential with two non-degnerate minima. Then there exists S > 0 such that
"H2W (h) - l + ofexp-fV \ hj
(5.9.7)
Note that this is not enough for getting the asymptotic behavior of the correlations.
5.10
Notes
Section 5.2 • There is another way to analyze the spectrum of the harmonic Kac operator by recognizing its kernel as the kernel of the operator exp-t(-d2/dx2 + ex2) (for some t > 0 and c > 0). This is the so-called Mehler formula (cf Simon [Sim2] or [He9]). Section 5.3 • A more general theorem than Theorem 5.3.2 can be found in [He2] or [He9] (see also the original lemma in [HeSjl]). Section 5.4 • Although we have tried to have a reasonably selfcontained presentation, it may be easier to read what is done in the case of the Schrodinger operator [He2]. Remark 5.4.2 and Section 5.5 is just a small presentation of the theory. • We refer to [Ir] for the stable manifold Theorem for hyperbolic maps. See also [He9]. • Standard material on symplectic geometry may be found in [Ho]. See also [He9]. • A more detailed presentation of this part with control with respect to the dimension is done in [He9] and in the research papers [Hel4], [Hel5], [HeRa] and [Moe].
74
Semi-classical
analysis for the transfer
operators
Section 5.5 • The harmonic approximation for the Schrodinger operator was probably already known at the beginning of the Quantum Mechanics. For rigorous proofs, one can refer to the books [CFKS], [He2], [He9] or [HiSig] or to the original papers [HeSjl] or [Sim5]. Let us also mention former contributions by Combes-Duclos-Seiler [CDS] and V.P. Maslov [Mas] (Appendix B and references therein). Section 5.6 • In the presentation of the upper bounds for the largest eigenvalue for the transfer operator, we have adapted the presentation of [CFKS] by proving something equivalent to the so-called IMS formula. The original proof was presented in [Hel4] but the harmonic approximation for the transfer operator is present in the course by M. Kac [Ka2]. Other techniques are present in for example [HeSjl] and (with control with respect to the dimension) [Sj5]. Section 5.7 • These results are of course very rough. Note that in the case when v is strictly convex, one has a universal lower bound for the splitting[He8], [HelO]. This extends a corresponding result for the Schrodinger operator due to J. Sjostrand [Sj6] (See also [HeSj3]) who was inspired by a Maximum principle argument given by [SWYY]. Section 5.8 • The role of the decay estimates was emphasized by B. Simon [Sim5] and B. Helffer-J. Sjostrand [HeSjl] in the case of the Schrodinger operator. But these estimates were present in a close context (decay at oo in the work by S. Agmon (and also before by Lithner) [Ag]. This is why we call them Agmon estimates. • The original results presented in this section and concerning this transfer operator come originally from [Bodl], [Hel4], [Hel5] and [Hel6]. • Another semi-classical model which is not analyzed here corresponds to the analysis of the operator exp —-j exp h2A exp — ^ . The analysis of this model was motivated by some question of
Notes
75
M. Kac and was analyzed in a series of articles [BruHe], [He4], [Hel4], [Hel5]. Here the eikonal equation we met in the construction of the WKB solution is the same as for the Schrodinger operator — h2A + V. This is not too surprising, as first observed by M. Kac, if one thinks of the Trotter-Kato formula [HelO], [Hell]. Section 5.9 • The strictly convex case is solved by [HelO] as already mentioned. • It is worth to recall that the papers of M. Kac and his collaborators [Tho] were probably at the origin of the interest of the mathematicians for the semi-classical analysis of the double well problem. We use also developments of the M. Kac's idea [Kal], [Ka2]. There are different variants of the so-called Hering formula which permit to analyze the splitting. The exact behavior of the correlations is given in [Hel6]. All the material presented here follows the ideas which were used in the Schrodinger case combined with the techniques of Kac and collaborators starting from [KaHel]. • In Subsection 5.9.2, we imitate the discussion given by HelfferSjostrand [HeSjl] for the analysis of the splitting in the analysis of the Schrodinger operator.
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Chapter 6
Basic facts in spectral theory and on the Schrodinger operator
6.1
Introduction
We summarize without proof the properties which are needed in spectral theory for the comprehension of this book with a particular emphasis on the Schrodinger operator. We illustrate in particular the theory by a careful study of the harmonic oscillator and of the Witten Laplacians. 6.2
Selfadjoint operators, spectrum and spectral decomposition
We consider an Hilbert space H and a dense vector subspace Ho of H. An unbounded operator L is determined by a given domain D(L) = Ho and by an operator mapping Ho into H. The operator is called a closed operator if the graph of L is closed in HxH. The operator is called a closable operator if the closure of the graph is a graph. We can then define the closure of the operator by a limit procedure. If Ho = H, the closed graph Theorem says that a closed operator L is continuous. We shall say that L : Ho •-> H is a symmetric operator if it satisfies (Lu | v)u = (u | Lv)-u ,Vu,v G Ho • It is easy to see that a symmetric operator is closable. Let us now recall the definition of the adjoint of an unbounded operator. 77
78
Basic facts in spectral theory and on the Schrodinger
operator
If L is an unbounded operator whose domain D(L) is dense in %, we define the adjoint operator L* by D{L*) = {ueH,D(L) 9 « i 4 ( u | Lv), can be extended as an antilinear continuous form on H} . Using the Riesz Theorem, there exists / G % such that (/ | v) = (u | Lv) , Vu G D(L*),Vv e D(L) and we can then define L* by L*« = / . It is then easy to see that L* is a closed operator. If L is symmetric it is easy to see that D(L) c D{L*)
(6.2.1)
Lu = L*u , Vu G D(L) .
(6.2.2)
and that
The two conditions (6.2.1) and (6.2.2) express the property that (L*, D{L*)) is an extension of (L, D{L)). We shall say that L is a selfadjoint operator if L* = L, i.e. D{L) = D(L*) , and Lu = L*u ,
Vu G D(L) .
For a symmetric operator, we have consequently two natural closed extensions : • the minimal extension is denoted by Lmin which is obtained by taking the closure of the graph of L. • the maximal extension is denoted by Lmax the adjoint of L. If Lsa is a selfadjoint extension of L, then Lsa is automatically an extension of Lmin a n d admits as an extension Lmax. A natural way to verify the uniqueness of the selfadjoint extension (see Section 6.4) is to prove the equality between Lmin and Lmax. Once we have a selfadjoint operator we can apply the basic spectral decomposition Theorem, which we now roughly explain.
Selfadjoint
operators, spectrum and spectral
decomposition
79
Definition 6.2.1 A family of projectors E(X), — oo < A < oo in an Hilbert space H is called a resolution of the identity if it satisfies : E(\)E(fi) E(-oo)
= £(min(A, n)) ,
(6.2.3)
= 0, E(+oo) = I,
(6.2.4)
where J5(±oo) is defined * by E(±oo)x =
lim
E(X)x
A—>±oo
for all x in H.
E(X + 0) = E(X) ,
(6.2.5)
where E(X + 0) is defined by E(X + 0)x =
lim
E{IJL)X .
We now observe that for all x and y in % the function A M- (E(X)x \ y) is with bounded variation and one can then show the existence of E{X + 0) and E(X — 0). It is then classical (Stieltjes integrals) that one can define for any continuous complex valued function A i-> /(A) the integrals Ja f(X)d(E(X)x | y) as a limit* of Riemann sums. The theory works more generally for any borelian function. We can then define an unbounded operator Lj whose domain is defined by D(Lf)
= L
|/(A)| 2 d(£(A)a; | x) < oo j ,
G U, f
and r+oo i-oo
/
f(X)d(E(X)x
| y)
-oo
for all x in D(Lf) and y in H. * (6.2.3) gives the existence of the limit. We observe indeed that A i-+ {E(\)x \\E(X)x\\2 is monotonically increasing. ^The best is to first consider the case when x = y.
\ x) =
80
Basic facts in spectral theory and on the Schrodinger
operator
We observe that for / 0 = 1 we have L/ 0 = I and for /i(A) = A, we obtain a selfadjoint operator Lf1 := L. In this case we say that +oo
/
A dE(\) -oo
is a spectral decomposition of L and we shall note that +OO
/-+O0
x\\2
/
X2d(E(X)x\x)
=
-oo
/
\2d\\E{\)
J—oo
for x G D{L). More generally •TOO
/-TOO
/ \f(X)\2d(E(X)x -oo
|/(A)| 2 d(||£;(A)x|| 2 )
\x)= J—oo
for x G D(Lf). The spectral decomposition theorem explicits that the preceding situation is the general one. Theorem 6.2.2 spectral decomposition Theorem Any selfadjoint operator L in an Hilbert space ~H admits a spectral decomposition. We now recall that the spectrum of a closed operator L is defined as the complementary set * of the set of the A G C such that (L — A) is invertible from D(L) onto H and admits a continuous inverse. The spectrum cr(L) is closed in C and real if L is selfadjoint. Note the following characterization. Proposition 6.2.3
a(L) = {A G M, s.t. Ve > 0, E{}\ - e, A + e[) + 0} . ^The complementary of the spectrum of an unbounded L is usually called the resolvent set. It is more precisely described as the set of the A in C such that the range of (L — A) is dense in "H and such that (L — A) admits a continuous inverse from Im (L — A) into D{L). If L is closed this implies that Im (L — A) = H.
Selfadjoint
operators, spectrum and spectral
decomposition
81
Applications of the spectral theorem One of the first applications of the spectral theorem is the following property which plays a crucial role and was already presented as Theorem 5.3.2 in Chapter 5. Theorem 6.2.4 d(X,a(L))\\x\\<\\(L-\)x\\,
(6.2.6)
for all x in D{L). This theorem is frequently used in the following context. It is usually difficult to get explicitely the values of the eigenvalues of an operator. One consequently tries to localize these eigenvalues by using approximations. Let us suppose for example that one has found Ao and y in D(L) such that ||(i-Ao)i/||<e
(6.2.7)
and 11 j/11 = 1 then we deduce the existence of A in the spectrum of L such that | A — Ao | < £• Standard examples are the case of hermitian matrices or the case of the anharmonic oscillator L := —h2-^ + x2 + x4. In the second case the first eigenfunction of the harmonic oscillator —h2-^ + x2 can be used as approximate eigenfunction y in (6.2.7) with Ao = h. We then find (6.2.7) with E = 0(h2). Another application is, using the property that the spectrum is real, the following inequality |ImA|||i||<||(L-A)i||
(6.2.8)
and this gives an upper bound on the norm of (L — A ) - 1 in C(H) by l/|ImA|. Examples of functions of a selfadjoint operator We shall for example meet usually in spectral theory the following examples of functions / : (1) / is the characteristic function of ] — oo, A], Yj_ooA] ; f(L) is then Lf = E(X).
82
Basic facts in spectral theory and on the Schrodinger
operator
f is the characteristic function of ] — oo, A[, Yj-oo^ ; f(L) is then Lf = E(X-0). f is a compactly supported continuous function. f(L) will be an operator whose spectrum is localized in the support of / . /((A) = exp(ifA) with t real. ft(L) is then a solution of the functional equation (dt-iL)(f(t,L)) f(0,L)
=0 =Id.
We note here that, for all real t, ft(L) = exp(itL) is a bounded unitary operator. <7t(A) = exp(—£A) with t real positive. gt(L) is then a solution of the functional equation (dt + L)(g(t,L))=0, 9(0, L) =Id.
ioit>0
Compact operators These operators were already considered in Chapter 4. In this case, all the theory can be made by diagonalization on a basis of orthonormal eigenvectors. Operators with compact resolvent We know that, if the operator L (which will be assumed left semi-bounded) is with compact resolvent, then its spectrum is a sequence of eigenvalues Xj tending to oo (each eigenvalue being with finite multiplicity). There are two conventions for the notation of the eigenvalues. The first one is to classify them into an increasing sequence Xj <
Xj+i
counting each eigenvalue according to its multiplicity. The second one is to describe them as a strictly increasing sequence Uk with fXk eigenvalue of multiplicity m*,. Examples (1) The Dirichlet problem in an open bounded set Q. One can for example explicitely compute the spectrum of the Dirichlet realization of a Laplacian in a square.
Discrete spectrum, essential
spectrum
83
(2) Laplace Beltrami operators on a Riemannian manifold. The simplest model is the operator —d2/d02 on the circle of radius one whose spectrum is {2im2 , n s IN}. For n > 0 the multiplicity is 2. An orthonormal basis is given by the functions 6 M- (27r)_2 expinO for n e Z5. (3) The harmonic oscillator on JR n . In this case a description of the eigenvalues and of the corresponding eigenvectors by the Hermite functions is standard (see 5.5 when n = 1). For the description of the spectral decomposition, it is better to take the second convention but we shall most of the time use the first one. If IIMfc denotes the projector on the spectral space attached to the eigenvalue fj,k of dimension mk, one can then describe E(\) as
The measure d (E(X)x \ y) is then equal to
d(E(\)x\y)=Y,^-»k)-(niikx\y). k
We can also write f(L) as
/(L) = ^ / ( M - n w . k
The other case to understand for the beginner is of course the case of the free Laplacian —A on Mn.
6.3
Discrete spectrum, essential spectrum
We have already recalled in Proposition 6.2.3 a characterization of the spectrum. Let us now complete this characterization by introducing different spectra. Definition 6.3.1 If L is a selfadjoint operator, we shall call discrete spectrum of L the set 0, dim range (E(] A - e, A + e[)) < +00} .
84
Basic facts in spectral theory and on the Schrodinger
operator
With this new definition, we get that, for a selfadjoint operator with compact resolvent, the spectrum is reduced to the discrete spectrum. Equivalently, we shall observe that A is in the discrete spectrum if and only if A is an isolated point in o~(L) and if A is an eigenvalue of finite multiplicity. Definition 6.3.2 The essential spectrum is by definition the complementary of the discrete spectrum. Intuitively, a point of the essential spectrum corresponds • either to a point in the continuous spectrum, • or to a limit point of a sequence of eigenvalues with finite multiplicity, • or to an eigenvalue of infinite multiplicity. Basic examples (1) The Laplacian on Mn —A is a selfadjoint operator on L2(Mn) whose domain is the Sobolev space H2(Mn). The spectrum is continuous and equal to 2R+. The essential spectrum is also M+ and the operator has no discrete spectrum. (2) The Schrodinger operator with constant magnetic field in 2R2:
with DXj := -dx 3
i
3
= T^— . i axj
The spectrum is formed with eigenvalues (2k + 1)\B\ but the spectrum is not discrete because each eigenvalue is with infinite multiplicity. It is clear that the essential spectrum is a closed set. In order to determine the essential spectrum it is useful to have theorems proving the invariance by perturbation. The following characterization is in this spirit quite useful. Theorem 6.3.3 Let L be a selfadjoint operator. Then A belongs to the essential spectrum if
Discrete spectrum, essential
spectrum
85
and only if there exists a sequence un in D(L) with \\un\\ = 1 such that un tends weakhfi to 0 and such that \\{L — A)u„|| —>• 0 o s n - > o o . Example 6.3.4 The operator — h2A + V with V tending to 0 as \x\ —> oo (a; e JR.0) has M+ as essential spectrum. For proving the inclusion of M+ in the essential spectrum, we can indeed consider the sequence u„(x) = n " ( D / 2 ) • exp(ix4)
• x((x ~
Rn)/n)
with x > 0 a n d supported in the ball 5(0,1) and equal to one on say 5(0, | ) . The sequence Rn is chosen such that \Rn\ tends to oo and such that the support of the un are disjoints. The converse can be obtained by abstract analysis and the fact that we know the essential spectrum of —A. Another way to localize the essential spectrum is to use the relative compacity. If L is a closed operator with a dense domain DT,, we shall say that the operator V is relatively compact with respect to L or L-compact if Dv C DL and if the image by V of a closed ball in DT, (for the graph-norm u i-> \/||w|| 2 + ||£w|| 2 ) 1S relatively compact in Ti. In other words, we shall say that V is L-compact, if, from each sequence un in DL bounded in % and such that Lun is bounded in H, one can extract a subsequence uni such that Vuni is convergent in ri. The Weyl's Theorem says : Theorem 6.3.5 Let L be a self adjoint operator, and V be symmetric and L-compact, then L + V is self adjoint and the essential spectrum of L + V is equal to the essential spectrum of L. The first part will be discussed in the next section. For the second part we can use the preceeding criterion. §We recall that we say that a sequence un in a separable Hilbert space ~H is weakly convergent if, for any g in "H, (un | g)fi is convergent. In this case, there exists a unique / such that (un \ g >-u~Jf (/1 g)-n and ||un|| is a bounded sequence.
86
6.4
Basic facts in spectral theory and on the Schrodinger
operator
Essentially selfadjoint operators
In most of the examples which were presented, the abstract operators are associated with differential operators. These differential operators are naturally denned on Co°(fi) or D'(fi). Most of the time (for suitable potentials increasing slowly at oo) they are also defined (when fi = Mm) on S(IRm) or S'(lRm). It is important to understand how the abstract point of view can be related to the PDE point of view. The theory of the essential selfadjointness gives a clear understanding of the problem. The question is to decide if, starting from a symmetric operator L, whose domain D(L) = Ho is dense in H, there exists a unique selfadjoint extension Lext of L (that is D(L) C D{Lext) and Lextu = Lu , Vu e D[L)). This leads to : Definition 6.4.1 A symmetric operator L with domain H0 is called an essentially selfadjoint operator if it admits a unique selfadjoint extension as unbounded operator onH. Examples (1) The differential operator - A with domain Ua = C^°(iR m ) is essentially selfadjoint and admits consequently a unique selfadjoint extension Lext whose domain is the Sobolev space H2{Mm) and which is simply denned as Lextu = -Au , Vu € H2 , where the r.h.s is denned in the sense of the distribution theory. (2) The differential operators - A + \x\2 with domain Ho = C^°(JR m ) or Hi = S(lRm) are essentially selfadjoint and admit consequently a unique selfadjoint extension. This extension is the same for the two operators. The domain can be explicitely described as B2(Mm) = {u e L2(Mm)
| xaD%u e L2{Mm)
, Va,p € W m w i t h | a | + |/?| < 2}
(3) The differential operator —A with domain Co°(fi) (where f2 is an open bounded set with smooth boundary) is not essentially selfadjoint. There exists a lot of selfadjoint extensions related to the choice of a boundary problem. We just mention here
Essentially
selfadjoint
operators
87
• the Dirichlet realization whose domain is HQ(CI) n H2(Cl) (where H% is the closure ^ of C£°(fi) in H1^)), • the Neumann realization whose domain" is
{u e # 2 (tt) I {du/dv)/dn = 0} (4) The Laplace Beltrami operator on a compact manifold M with domain C°°(M) is essentially selfadjoint on L2(M). The domain of the selfadjoint extension can be described as H2(M) (which can be described using local charts). We now give some criteria in order to verify that an operator is essentially selfadjoint. As already mentioned, one can prove essential selfadjointness by proving that the minimal closed extension coincides with the maximal closed extension : J-tmin — ^ m o i •
Here we recall that the minimal extension of a symmetric operator L of domain "Ho is the operator £ m ,„ whose graph is the closure of the graph of L. Lmax can be seen as the adjoint operator L* and Lmin as L**. The second criterion is to establish the injectivity of Lmax ±i. We have indeed the following general criterion. Theorem 6.4.2 Let A be a closed symmetric operator. Then the following statements are equivalent : • A is selfadjoint. • ker (A* ±i) = {0} ; • Range (A±i) = H. This theorem gives a criterion for essential selfadjointness in the form : Corollary 6.4.3 Let A with domain D(A) be a symmetric operator. Then the closure of A A** is selfadjoint if and only if ker (A* ± i) = {0}. ' W e recall that if u € H1 (CI) the restriction of u to dCl is well defined and belongs to the Sobolev space H^(dCl). The space H^(Cl) can also be characterized as H^(Cl) =
{ueH1(Cl)s.t.u/an=0}. II We recall that if u £ H2(Cl) the restriction of du/dv to the Sobolev space H5 (dCl).
to dCl is well defined and belongs
88
Basic facts in spectral theory and on the Schrodinger
operator
We shall see in the next section how to apply it in simple cases. In the same spirit, we have in the semibounded case the following : Theorem 6.4.4 Let A be a closed, positive, symmetric operator. Then A is selfadjoint if and only if ker (A* + b) = {0} for some b > 0. Similarly, if A is a positive symmetric operator, then the closure of A, A** is selfadjoint if and only if this condition holds. Remark 6.4.5 Another way to verify the condition is to show that Range (A + b) is dense.
6.5 6.5.1
Examples The free
Laplacian
In this spirit let us treat first LQ = — A with domain 'HQ = Co°(JR m ). It is immediate to verify that —A is symmetric (we say also formally selfadjoint) and we recall that C§°{Rm) is dense in U = L2(Mm). The domain of the smallest closed extension appears as the subset of the «'s in L2{Mm) such that there exists a sequence Uj G Co°(2Rm) such that Uj —> u in L2 and such that — AUJ converges in Li2. We then define Lminu as limj_).00 — AUJ. The theory of distributions says actually that Lminu = —Aw in the sense of the distribution theory. The adjoint L* = Lmax (which is also equal to L^in) is by definition the operator whose domain is D(Lmax)
= {ue L2(Mm)
| - Au e
L2(Mm)}
and which is defined by D{Lmax)
3 u H» - A u .
Here we have used the Riesz Theorem, characterizing the dual of L2. It is clear that Lmin = Lmax using easy Fourier analysis. It is sufficient indeed to prove a result of density, that is to prove that if {u £ L2 , —Au € L2} then there exists a sequence Uj in CQ° such that Uj —¥ u in L2 and such that — AUJ converges in L2. If one first recognizes D(Lmax) as H2 (which is immediate via Fourier), this is simply a consequence of the density of C%°(ST) in H2(IRm).
89
Examples
Of course this example is in some sense too simple for seeing really the problems which may occur. 6.5.2
The harmonic
oscillator
As we have already seen, the harmonic oscillator Hh =
-h2d2/dx2+x2
plays a central role in the theory. We recall that its spectrum is given by a sequence of eigenvalues Xn(h) (n e IN*) Xn(h) =
(2n-l)h.
In particular the fundamental level is X1(h)=h and the splitting between the two first eigenvalues is 2h. The first eigenfunction is given by x2
_i
<j>\{x,h) =cih
4ex
P~7w-
and the other eigenvectors are obtained by applying the creation operator L+ = —hdx + x . We observe indeed that Hh = L+ • L~ + h where L~ = hdx + x = (L+)* and has the property L-<j>i = 0.
The n t/l -eigenfunction (which is an Hermite function) is then given by K = cn(L+)n-Vi • Before to treat this example, let us first mention the Kato Theorem :
90
Basic facts in spectral theory and on the Schrodinger
operator
Theorem 6.5.1 Let V in L2oc(Mm) such that V > 0 almost everywhere on Mm. —A + V with domain Co°(iR m ) is essentially self adjoint.
Then
We now consider the case V = \x\2. Of course the above criterion can be applied. A pseudodifferential proof ** can also be given which will show that the domain of the operator is B2(Rm)
= {ue L2{Mm)
| xaD%u G L2(mm),\/(a,(3)
with \a\ + \f3\ < 2 } .
Let us sketch, a more elementary proof. Following the scheme of the proof given for the free Laplacian, it is sufficient to prove that if u G L2 and (—A + V)u G L2 then u G B2 and we then use the well known density of C§° in B2. We first remark that the local regularity theorems concerning elliptic operators give that u G H2oc. We then obtain for a suitable cutoff sequence Xn Re ((-A + V)u | Xnu) = Jxn\Vu\2dx
+ Jv.Xn\u\2dx-±[(Axn)\u\2
.
This gives that u € B1 and a meaning to Re ( ( - A + V)u\u)=
J \Vu\2dx + f V • \u\2dx .
In order to get the regularity in B2 we can use the method of the differential quotients, that is apply the inequality to the family ——f u (where r^ e . is the translation by hej with e^, j—th vector of an orthonormal basis in Mm) and to the family 2 2 E i ^ i u .
Remark 6.5.2 What is claimed here for the harmonic oscillator can be extended to C°° " L e t us consider the harmonic oscillator as a pseudodifferential operator p(x, Dx) whose symbol i s p ( x , £ ) = £ 2 +x2. Consider the pseudodifferential operator q(x,Dx) whose symbol is
q(x,0 = l/(x2+e
+ l)
Then q(x, D) o p(x, D) = Id + r(x, D) and r(x, D) and q(x, D) can be seen as continuous from Bk into Bk+2
for any k.
More on
selfadjointness
91
potentials V satisfying in IRm
V(x)>±\x\2-C, for some constant C > 0, and \D°V(x)\
6.6
<
Ca(x)V-W+.
More o n selfadjointness
We first recall the following general theorem. Theorem 6.6.1 A symmetric semibounded operator L on H (with D(L) dense in H) admits a selfadjoint extension. The standard extension appearing in the proof is the so-called Friedrichs extension. This is a variant of Lax-Milgram Lemma. We can assume indeed by adding possibly a constant that L is strictly positive and consider the associated form a priori defined on D(L)"2 (u,v) H-> #L(U, I;) = (Lu \ v). We call V the completion of D(L) for the norm u H-> 5Z,(U, U). By Lax-Milgram Lemma, we get an operator C from V onto V, and we have the canonical sequence V C H C V. The selfadjoint extension L is given by the restriction of C to D(L) = {ueV
, CueH}
.
In addition let us mention the following statement. T h e o r e m 6.6.2 A Schrodinger operator on Mn with C°° potential which is semibounded on C£°(]Rn) is essentially selfadjoint. In other words, the Friedrichs extension is the unique selfadjoint extension starting from C^0(IRrn). This last theorem is complementary to Theorem 6.5.1 because we do not have to assume the semi-boundedness of the potential. Proof Let P our operator. Possibly adding a constant, we can assume that (Pu | u)H > INI 2 •
(6.6.1)
92
Basic facts in spectral theory and on the Schrodinger
operator
According to the general criterion of essential selfadjointness (cf Remark 6.4.5), it is enough to verify that R(P) is dense. Let us show this property. Let / G L2(Mm), such that (/ | Pu)n
= 0 , V« € C0°°(iRm) .
(6.6.2)
We have to show that / = 0. Because P is real, one can assume that / is real. We first observe that ( - A + V ) / = 0 i n V'{Mm). A standard regularity theorem implies that / € C°°{IRm). We now introduce a family of cutoff functions £& by Cfc := C(x/k) , Vfc e IN ,
(6.6.3)
where ( is a C°° function satisfying 0 < £ < 1, £ = 1 on B(0,1) and supp CC 5(0,2). We obtain (V(C*/) I V(C*/)> + J ekV(x)\f(x)\2
dx = J | V a | 2 | / ( x ) | 2 d x .
(6.6.4)
Using (6.6.2) and taking the limit k —>• +00, we get ||/||2=
lim | | a / H 2 < (PCkf\Ckf)= fc—>+CO
lim
[f(x)2\(Va)(x)\2dx
= 0.
K—V+OO J
(6.6.5) This proves the theorem. As a corollary of this theorem, we get immediately that : Corollary 6.6.3 All the Witten Laplacians (corresponding to phase in C°°) are essentially selfadjoint. Proof They are indeed positive. Example 6.6.4 The realization of the Dirichlet problem for the Laplacian in an open set ft is obtained by this way. In the case of the Schrodinger operator with a bounded from below C°° potential on Cl, the domain D of the extension
The max-min principle
93
will be described as D = {u G f#(fi) , (V + C ) ^ £ L2 and ( - A + V)u € L 2 } . They are many cases where one can prove a regularity theorem giving that if u e D then - A u € L 2 and ^ u £ L 2 .
6.7
The max-min principle
We now give two criteria for the determination of the bottom of the spectrum and for the bottom of the essential spectrum. Theorem 6.7.1 Let A be a selfadjoint semibounded operator and Q(A) its form domain. ^ Let us introduce /i n (A) = ^
max *-1
min (A(f> | (f>) . f > € [apar.tyi l ...,Vn-i)]- L ;l
\<j)£Q{A) and ||0|| = 1
J
Then either (a) fJ,n(A) is the n-th eigenvalue when ordering the eigenvalues in increasing order (and counting the multiplicity) and A has a discrete spectrum in ] - oo, fin(A)] or (b) fin{A) corresponds to the bottom of the essential spectrum. In this case we have Hj(A) — fin(A) for j >n. Remarks • It is very often useful to apply the max-min by taking the minimum over a dense set in Q(A). • For the first eigenvalue the criterion gives fH(A) =
inf
(A | <j>) .
eQ(A) and ||0||=i
• The max-min principle permits to control the continuity of the eigenvalues with respect to parameters. ^Associated by completion with the form u i-> (u|ylu) initially defined on
D(A).
94
Basic facts in spectral theory and on the Schrodinger
operator
• The max-min permits to give an upperbound on the bottom of the spectrum and the comparison between the spectrum of two operators. • In the case when A = - A + V on Mm : a(A) C [infV,+oo[.
6.8
(6.7.1)
Compactness
We have already discussed the notion of compactness in the preceeding chapter. We recall that an operator T is a compact operator if the range of the unit ball B of H by T is relatively compact in H. In the Hilbertian context, another characterization is that T is compact if from each weakly convergent sequence (xn)n€N G Ti one can extract a subsequence (xni)iew, such that the sequence (Txni)i€N is strongly convergent. We shall denote by C{7-L) the subspace of the compact operators. Let us recall the following properties : • if T G C{H) and A G £(71), then TA G C{U) and TA G C(U). • if T G C(U)), then T* eC{U). We just recall some criteria of compactness for the resolvent of the Schrodinger operator P = - A + V on Mm. If V is C°° and bounded from below, the domain of the selfadjoint extension is always contained in the form domain that is Q{P) := H^{IRm)
= {u€ Hl{Mm)\{V
+ C)*u G L2(Rm)}
.
It is then easy to see that if V tends to oo then the injection of Hy in L2 is compact (use a criterion of precompactness). We then obtain that the resolvent (P — A ) - 1 is compact for A §?• cr(P). In the case of a compact manifold M and if we consider the LaplaceBeltrami operator on M, then the compactness of the resolvent is obtain without additional assumption on V. The domain of the operator is H2(M) and we have compact injection from H2(M) into L2(M). The condition that V —>• oo as |a:| —> oo is not a necessary condition. The Schrodinger operator on M2: —A + x2-y2is with compact resolvent (see exercise 6.8.3).
95
Compactness
T h e case of t h e W i t t e n Laplacian Let us consider the Witten Laplacian A$ (2.3.2), it has the form
on 2Rm. As we have seen in
Ai 0 ) = - A + i | V $ | 2 - ^ A $ .
(6.8.1)
So a first natural sufficient condition for compactness of the resolvent is the condition V(x) := - | V $ ( z ) | 2 - - A $ ( z ) -> +oo , as |z| -> +oo .
(6.8.2)
We first give a computation showing that this condition is not too far from optimality. We can indeed prove : L e m m a 6.8.1 Suppose that V > 0 and that there exists r > 0 and a sequence o~n such that \o~n\ —> +oo and such that sup
sup
V(T) < +oo .
(6.8.3)
nelVreB(
Then the self-adjoint realization of — A + V is not with compact resolvent. Proof Let us consider the sequence 4>n(x) = ip(x - a„) .
(6.8.4)
Here ijj is a compactly supported function of L2 norm 1 and with support in 5 ( 0 , r ) . We observe that the <j>n are an orthogonal sequence (after possibly extracting a subsequence) which satisfies for some constant C ll^||^+||^n|||2
(6.8.5)
We can not extract from this sequence a strongly convergent sequence in 1?. On the other hand the sufficient condition does not look optimal. We have indeed the following proposition. Proposition 6.8.2 Let us assume that there exists 9 €]0,1[ such that 0|V$(a;)| 2 - A$(z) -»• +oo , as \x\ -> +oo .
(6.8.6)
96
Basic facts in spectral theory and on the Schrodinger
operator
Then the Witten Laplacian A<j,' is with compact resolvent. We just explain the case m = 1. We have bf=Z*Z,
with Z = ^
+ \'{x) .
(6.8.7)
We now observe that Z + Z* = '{x) ,
(6.8.8)
[Z, Z*\ = "(x) .
(6.8.9)
and
We now write, for any a > 0, \\(Z + Z*)u\\2 = \\Zu\\2 + \\Z*u\\2 + (Zu | Z*u) + {Z*u | Zu) < ( l + a)||Z U || 2 + (l + i ) | | ^ U | | 2 -
{
'
We now implement the property that \\Z*u\\2 = \\Zu\\2 + I'(j>"{x)\u{x)\2dx .
(6.8.11)
This gives the inequality f('(x)2 - (1 + -)" (x))\u(x)\2dx
< (2 + a + -)(Aj, 0 ) U | U)L2 , (6.8.12)
and the compactness of the resolvent. Exercise 6.8.3 Show that the unbounded operator on L2(M2) P-=-—
- — dx dy2 2
x22
is with compact resolvent. Hint One can introduce ^ i = ~9X , X2 = -dy , X3 = xy , i
i
97
Notes
and show, for j — 1, 2 and for a suitable constant C, the following inequality : \\(x2 + y2 + 1 ) - ^ , X 3 H | 2 < C {{Pu | u)L*{m
+ ||u|| 2 ) ,
for all u e C%°(1R2). One can also observe that i[X1,X3]=y,
i[X2,X3}=x,
and then ll^a +y2 + 1)-i[Xl,X3]u\\2 >\\(x2+y2 + l)iu\\2.
+ \\(x2 + y2 + l)-i[X2,X3}u\\2
+ \\u\\2
For the control of ||(x 2 + y2 + l)~i [Xu X3] u\\2, one can remark that IK*2 + y2 + 1 ) - * [ X 1 ; X 3 H | 2 = {. „ \ ^ u [xz +y2 + 1)4 and perform an integration by parts.
| (X,X3
- X3X!)u)
,
Exercise 6.8.4 Show that if m = 2 and $ = (a;2 + e)(y2 + e) with e > 0, then A^0) on L2(M ) is with compact resolvent. We emphasize that in this case, for e > 0 small enough, Proposition 6.8.2 can not be applied. 6.9
Notes
Section 6.1 • There are many books where this material is presented : [CFKS] [Hel], [Hu], [RS-I] - [RS-IV], [Rob2]... [HiSig] to mention some. Section 6.2 • For the basic "spectral decomposition Theorem", we refer to K. Yosida [Yosi] (p. 309-316) or to the book by D. Huet [Hu]. • For the discussion around the characterization of the spectrum (see footnote), we refer to [Yosi] (p. 209). • For Proposition 6.2.3, we refer to appendix 2 in the book of D. Robert [Rob2].
98
Basic facts in spectral theory and on the Schrodinger
operator
Section 6.5 • The analysis of the harmonic oscillator was of course the guide line for the analysis of the harmonic Kac operator analyzed in Chapter 5. • The Kato Theorem can be found in [RS-IV]. • For the presentation of pseudo-differential techniques, we refer to [Ho], [Hel], [Rob2] and in the context of the Witten Laplacians to [Jo]. Note that these techniques are not necessary for rough results but may be quite useful for refined results. • The differential quotients method is a standard method which can be found in [LiMag]. Section 6.6 • The construction of the Friedrichs extension can be found in [Yosi], p. 317. • For this theorem, which answers to an old conjecture, we refer to the paper of Simader [Sima] and references therein. This is treated also in the book of G. Royer [RoyG2], who refers to [Wie]. • The analysis of the maximal domain of a Schrodinger operator is a large subject for which we refer to [Robl]. Section 6.7 • We refer to [RS-III] for these criteria. Section 6.8 • For more general criteria concerning the compactness for resolvents of Schrodinger operators we refer to [HeMo] and references therein. • Proposition 6.8.2 is a particular case of [BolDaHe], which was motivated by a question of F. Mignot. This is also discussed in [Jo]. • Lemma 6.8.1 is taken from an exercise presented in [DeStl]. • For the spectral theory of the Laplace-Beltrami operator on a compact manifold, we refer to the book of Berger-Gauduchon-Mazet [BGM]. • Exercise 6.8.3 corresponds to some example of [HeMo]. There are many other proofs for treating this example, see for example B. Simon [Sim3] and the book with J. Nourrigat [HeNo].
Notes
99
• The case when e = 0 in Exercise 6.8.4 is also interesting. We conjecture that in this case, this is not with compact resolvent, but the problem is open. We thank F. Nier for useful discusssions around this point.
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Chapter 7
Log-Sobolev inequalities
7.1
Introduction
We recall from the introduction that we are interested in the following inequality. We consider a measure* dfi := exp—$ dX on M = Mm and want to analyze the following inequality, which is called the log-Sobolev inequality : /
|/|2ln|/|2dM-||/||2ln||/||2
JM
/
|V/|2^.
(7.1.1)
JM
We shall assume that
/
dii = \ .
(7.1.2)
M Jh
The best constant such that (7.2.2) is satisfied (if it exists) will be denoted by CL.s.. This inequality can be seen as a control of the entropy, which is defined, for a non positive function g such that E^.(g\n+ g) < +oo, in the following way : EnVfo) = Edging)
- E^g) InE^g)
.
(7.1.3)
Here we have used the standard notation :
E^(g) = (g)ll = j gdpi "On a compact riemannian manifold M, this would be a measure of the type exp — da where da is the riemannian measure on M. 101
102
Log-Sobolev
inequalities
We note that the entropy has the following properties EntM(fl) > 0 ,
(7.1.4)
Ent»(ag) = a Ent^(g) , Va > 0 .
(7.1.5)
and
One of the interesting properties related to the log-Sobolev inequality is the property relative to the tensor product of two measures fii and \X2 defined respectively on two manifolds Mi and MiTheorem 7.1.1 / / we have the log-Sobolev inequality relative to L 2 ( M j ; dfii) and L2(M2 ; d/j.2), with respectively constants CL.S.(MI,HI) and Cz,.s.(-M2,/i2)j then we have it in L2(Mi x Mi; d(ni <E) ^2))- Moreover the LS constant satisfies CL.s.{MixM2,
m®H2) <max(CL.sXMi,m)
, CL.S.{M2,^))
. (7.1.6)
Proof Starting of a given function strictly positive / in C£°(Mi x M2), we first apply the log-Sobolev inequality with respect to the variable x\ £ Mi and get with obvious notations
/
l / l M n l / l 2 ^ ^ 11/11?In 11/11?+ C-! f
\Vif\2d„i.
(7.1.7)
Then one integrates with respect to X2 and we would like to control
/:= /
ll/llJlnll/Hfd^.
(7.1.8)
We apply the log-Sobolev inequality to the function g : M23x2^g(x2) = ( / M i f{xi,x2)2 d/xij 2 . The only point which is not clear immediately is the proof of /
|V 2 g{x2)\2dn2 < [
J Mi
|V2/(zi,z2)|2^i-^2.
(7.1.9)
JM1XM2
Here we observe that IM2 |V 2 9{x2)?dH2 = JM2 & |V 2 {92){x2)\2dlX2 = / M 2 F I SM1 f(xux2)(^2f)(xi,x2)dni\2dfi2
• (7.1.10)
Log-Sobolev inequalities in the strictly convex case
We can then apply Cauchy-Schwarz in the xi variable.
103
#
This property makes log-Sobolev inequality interesting for the analysis of a large number of particles, that is for the case when we consider the manifold M^ = MN. One can indeed get an upper bound for M< N CL.S.( ~ \ M ® ' • • ® M) which is independent of N. 7.2
Log-Sobolev inequalities in the strictly convex case
We prove in this section the log-Sobolev inequality under a condition of uniform strict convexity of the phase $ defined on M = MN. More precisely we assume the existence of A > 0 such that Hess $(x) > A , Var € M .
(7.2.1)
T h e o r e m 7.2.1 Under the conditions (7.2.1) and (7.1.2), the following log-Sobolev inequality is satisfied : f |/|2ln|/|2^-||/||2ln||/||2<|
/
|V/| 2 rf/i,
(7.2.2)
with dfi := exp —$ dX. Consequently, the best log-Sobolev constant satisfies :
CL.S. < I •
(7.2.3)
Theorem 7.2.1 will be obtained as a particular case of the following more general : T h e o r e m 7.2.2 Under assumptions'1 (7.1.2) and (7.2.1), and if the function ^ belongs to C°([0, +oo[) nC°°(]0, +oo[) and satisfies the conditions : *" > 0 , 3C > 0 s. t. *"(t) < C , fort>C
(7.2.4) ,
(7.2.5)
t O u r proof uses also other technical assumptions which will be mentioned later on the derivatives of $ which are probably essentially technical.
104
Log-Sobolev
inequalities
and (1/*")" < 0 ,
(7.2.6)
then we have
f *(f)d» - * ( / " / ) < J - /" *»(/) |V/|2rfM , JM
JM
AA
(7.2.7)
JM
for any f > 0 in the class of the C 1 functions with bounded derivatives. The basic examples for which the assumptions (7.2.4) and (7.2.6) are satisfied are • ^f(x) = x2 and this leads to Poincare inequality, observing the inequality |V|/| | < | V / | a.e. ,
(7.2.8)
for / G H\ • ty{x) = a;Ins and this leads (after a change of functions f = g2) to the standard log-Sobolev inequality, • ^(x) = xp, with 1 < p < 2. The proof goes essentially as follows. Step 1 We begin with the following preliminary lemma : Lemma 7.2.3 It is enough to prove the theorem with the same property for $ but with the additional property that *eC°°([0,+oo[).
(7.2.9)
Proof We just observe that we can consider the family ^e defined, for e > 0, by * £ ( t ) = * ( * + £).
(7.2.10)
If (7.2.7) is satisfied for a given / , for the family ^fe, we can, by applying dominated convergence Theorem, take the limit e —• 0 in the inequality under the weak assumption that / satisfies J \f\2 dfi < +oo.
105
Log-Sobolev inequalities in the strictly convex case
W e now a s s u m e t h a t \I> satisfies the additional condition (7.2.9). Other preliminaries We denote by A$ the Laplacian associated with the Dirichlet form |V/| 2 d/x = ( 4 0 ) / | f)LHM,d»)
/ -> /
,
(7-2.11)
JM
and by Pt the associated semi-group Pt = e x p - ^ 4 0 ) , V i > 0 .
(7.2.12)
As we have mentioned, the operator A$' is always essentially selfadjoint on L2(M, dfi) starting from CQ°(M) and has 0 as lowest eigenvalue associated to the eigenvector x H-> 1. If we denote by A2 the second eigenvalue of A§', assuming for a moment that it is strictly positive eigenvalue (so 0 is a simple eigenvalue), we obtain by abstract analysis the following property \\Ptf-
I fd»\\h(M,dri<exp-l\2t JM
• | | / - [ fdn\\2La{Midlt)
z
. (7.2.13)
JM
Here it is probably useful to recognize that / K> {x H4 J fdn} ,
(7.2.14)
JM
is the projector on the first eigenfunction of .A^'. Although it is also a consequence of a general theorem for Schrodinger operators which we recall later, it is immediate to see that 0 is simple. We get indeed immediately the property that if u e D(A$'), the equation A)j/u = 0 implies, by taking the scalar product in L2(M,dfi), the equation du = 0 and consequently u = const. . So, under our conditions of strict convexity of $, the spectrum of A$' being discrete, 0 is simple.
106
Log-Sobolev
inequalities
Let us also note that Pt has the positivity preserving property, which means essentially* that it has the property Vv € C^(M)
, v > 0 implies Ptv > 0 pointwisely .
(7.2.15)
This means that Pt has the same property as the transfer operator. This property can be proved by using the Trotter-Kato formula. Theorem 7.2.4 Let A and B be selfadjoint operators on a separable Hilbert space so that A + B defined on D{A) C\ D(B), is selfadjoint. Let us assume that A and B are semibounded. Then / tR tA /R\n exp —t(A + B) = s — lim I exp exp exp I , (7.2.16) n-y+oo \ 2n n 2nJ where the limit is considered in the sense of the strong convergence. This property of positivity preservation being stable by strong limit, this gives the result if V = ^ | V $ | 2 — A $ is semi-bounded, that is if there exists C such that V{x) > -C .
(7.2.17)
This is at least the case when $ is of class C 2 with bounded derivatives of order 2. This is also the case for the toy models considering in our course. Remark 7.2.5 We actually have the stronger property saying that if v > 0 and v not identically 0, then Ptv > 0 for t > 0. This property is called positivity improving property and can be verified by using the Feynmann-Kac formula but this supposes a good knowledge of the properties of the measure. As a consequence of the Krein-Rutman Theorem 4.6.3 we get, for any selfadjoint Schrodinger operator —A + V with discrete spectrum, the following Theorem 7.2.6 For any t > 0, Pt is positivity improving. When V —> +oo ; Pt has as largest eigenvalue exp —| Ai as simple eigenvalue. \i is the simple eigenvalue of —A + V and there exists a strictly positive eigenvector. '•Actually, this is a little more. One needs to say, following Reed-Simon [RS-IV], that if v 6 L2, v ^ 0 and v > 0, then Pt v > 0 and not identically 0.
Log-Sobolev inequalities in the strictly convex case
107
Exercise 7.2.7 Show directly the strict positivity of the kernel of Pt (t > 0) when V(x) > •g\x\2 — C by comparing with the quadratic case. This ends our remark.
#
B u t let us come back t o t h e proof of t h e log-Sobolev inequality. Step 2 For a non negative function / £ CQ°(M),
let us consider
g{t) = [ *(P«/)d/z .
(7.2.18)
JM
We first have the following : Proposition 7.2.8 g(+oo) = I * ( / JM
JM
fdfi) d/x = ¥ ( /
fdfx) .
(7.2.19)
JM
Proof of Proposition 7.2.8 Although the statement is rather clear heuristically from (7.2.13), it is less easy to give a correct proof. We would like to use again dominated convergence Theorem. So we have to find a suitable upper bound for ty(Ptf) uniform with respect to t large and to show that ^f((Ptf)(x)) converges to ^(JM /^A4) a l m o s t everywhere as t -»• +oo. Let us control Ptf — JM f dfi uniformly. For a local control in x, it is enough to control this quantity in the Sobolev space HJ°oc(M; d/j.) and to use the Sobolev inequality. This control in the local Sobolev spaces is a consequence of a regularity theorem saying that £>((4 0) ) fc ) C Hf0kc(M; d/x) , Vfc € IN ,
(7.2.20)
which is nothing else than the traditional local regularity theorem for elliptic operators of second order. We need actually some stronger information which will also be a consequence of the control of the expression (A%')k (Ptf — JM f d/j.) in
108
Log-Sobolev
L2(M;
inequalities
d/j,). One has the following extension of (7.2.13)
\Uf)k{Ptf-SMfdiA\\L,{M,dtl) = ||(4 0) ) fc e x p - i 4 ° 5 (f-fMfdp)
\\v{M,M
(7-2.21)
In particular we get (via Sobolev inequality) the uniform convergence of Ptf to fM fdfi on any compact. This is not enough for completing the argument because we would like to use dominated convergence theorem. Coming back to the Witten-Laplacian point of view, we first state the following property for the Witten Laplacian A<j,' which will be admitted in this course : Theorem 7.2.9 Let us assume that $ satisfies the following conditions. • There exists C such that |V$(x)| > — < x >i
- C .
(7.2.22)
• For some p > 0, there exists, for any a, Ca > 0 such that : |D°V$(a:)| < Ca < V$(z) >(1-"IQD+ .
(7.2.23)
Then, for any £ £ IN, there exists k £ IN such that D((Aff)cB'(M), where Bl{M) B\M)
(7.2.24)
is defined by
= { « £ L 2 (2R m ) | xaDiu
e L2(Mm)
, Ma,(3 s. t. \a\ + |/3| < 1} . (7.2.25)
This is a standard result in the theory of the globally hypoelliptic operators. By using the unitary transformation from l?(M,d\x) on the standard L2 space, one gets the improvment that, for any I £ IN, there exists Ce such that : < x >e exp - ^ p
(Ptf{x)
- f
fdii\l,Vx£M. (7.2.26)
This gives us a control in the form *(Ptf(x))
< Ce exp$(a;) < x >~e .
(7.2.27)
Log-Sobolev inequalities in the strictly convex case
109
< s >2.
Here we have used the property that \£(s)
#
The second point, which is proved similarly, observing that 1i-»- Ptf is for / G nfc£>((4>0))fc) a C°° function from C°°(M+) into n f c D((^ 0 ) ) f c ). Proposition 7.2.10 When / > 0 and in C\kD((A^')k),
then we have
lim g(t) = 5 (0) = / <->o+
¥ ( / ) d/i .
(7.2.28)
JM
We finally obtain (after proving that g is differentiable) p
r
/
*(/)d/i - * ( /
7M
/"+oo
/d/i) = - /
JM
g'(t)dt
(7.2.29)
d» .
(7.2.30)
JO
with
/
A{°\Ptf)
*W)
We perform an integration by part (see (2.5.1)) and get 9'(t) = - \ ( L
*"(Ptf)(VPtf
| V P t / ) d>i.
(7.2.31)
JM
As a result of this second step, we have obtained : Proposition 7.2.11
/ *(/)dM - * (7 ^ = \ r ° ° i WW) (v^/1 vpt/> dMdt. JM
V/M
/
^ Jo
JM
(7.2.32) So the extended log-Sobolev inequality (7.2.7) is a consequence of the following inequality
r°° / *"(pt/)(vpt/1 vpt/) d^dt <\ f *"(/)(v/1 v/> d,i.
./o
A
JM
JM
(7.2.33)
Step 3 If we introduce K(t) = [ * " ( P t / ) ( V P t / | V P t / ) d/i , JM
(7.2.34)
110
Log-Sobolev
inequalities
the preceding equation can be rewritten as -t
+00
L
K(t)dt < - K{0) .
(7.2.35)
A
Let us observe that the convexity of $ implies K(t) > 0 .
(7.2.36)
The inequality (7.2.35) is of course a consequence of the integration between 0 and +00 of the inequality K{t) < exp -Xt • K(0) .
(7.2.37)
This last inequality is itself a consequence of K'{t)<-XK{t).
(7.2.38)
Step 4 Proof of (7.2.38) In order to analyze under which condition this inequality is satisfied we introduce the notations ip = V"
(7.2.39)
and Ptf(x)
= h(u(x)) .
(7.2.40)
We take h such that V'(h(s)) =Cs + C ,
(7.2.41)
with C > 0. This is well defined because ($?')' > 0 by assumption. Note that by differentiation of (7.2.41), we get, iP(h(s))h'{s) =Cy£0.
(7.2.42)
Example 7.2.12 If we first consider the example ^f(x) = x2, we get ^ ' ( x ) = 2x and in this case, with C = 1 and C = 0, h(s) = ^s. If we consider now the example ^(x) = x l n x , we get fy'(x) = lnx + 1 and in this case, with C = 1 and C — 1 the choice of h(s) = es. More generally, when fy(x) = xp, we get h(s) = s ? 7 1 .
Log-Sobolev inequalities in the strictly convex case
Using this change of function u = h^1(Ptf) (7.2.38) takes the form,
111
and (7.2.12), the inequation
2\fMi>(h(u(x)))h>(u(x)nVu(x)\\*d»
<j
M v>(M«(*)))(VM«)|v4
0)
(M«))>d/x
(7 2 {
+ fMTt,(h(u(x)))(V(Afh(u))\Vh(u))d» + $MiP'{h{u))h>{u)* {Afh{u)) \\Vu\\*dn . Here h{u) is the function x >->• /i(u(a;)) and V(A^
• •
43)
)
h(u)) means that one
applies V to the function x H-» (A^'(h(u(-))) J (a;). If we introduce the Witten Laplacian on the one-forms we can first rewrite 2\fM4,(h(u(x)))h>(u(x)nVu(x)\\2d» < 2 JM V'(/i(«( a; )))(V/i(u)|4 1) Vh(u))dn + fM(u)2 (A^h(u)) ||Vu|| 2 d/z-
(7.2.44)
We plan now to use the identity (already used in (2.5.2) ) : A{V = 4>0) ® I + Hess $ .
(7.2.45)
We first analyze the first term of the right hand side in (7.2.44) and get fM ^(hiuixMVhiu^Vhiu^dn = JM iP(h(u(x)))h'(u)2(Vu\ Hess $ Vu)efyi + JMi>(h(u(x)))(Vh(u)\(A<£)®I)Vh(u))d»
(7.2.46) .
JM ^(fc(u(x)))(Vft(u)|4 1) V/i(u)>d^ = JM iP(h(u(x)))h'(u)2(Vu\ Hess $ Vu)dji
(7.2.47)
+ Eij /*,( W ) M 0 2 ) ' ( « ) djU diUfl?.udp. Let us now treat the second term of the right hand side in (7.2.44). Let us first observe that Afh{u)
=
ti{u)Afu
- h"(u)(Vu
| Vu> .
(7.2.48)
112
Log-Sobolev
inequalities
We get, using (7.2.48) and integrating by parts for getting the second equality,
JM V{h{u))h'{u? 4 0 ) M U )||V U || 2 ^ = /M(^(M-))^(-)3)H(40)")l|vw||2 dp -!M^{h{u))h'{ufh"{u)\\Vu\\^ = IM V U • V [i>'(h(u))h'(u)3\\Vu\\2] dfi -fMi>'(h(u))h>(u)2h"(u) ||VW||4^
(7.2.49)
= JM W{h)h«)' - V{h)h>2h") (U) ||V U || 4 ^ JMiP'(h(u))h>(u)3Vu.V(\\Vu\\2)dn.
+
Let us concentrate for a while on the term [ ( V > W 3 ) ' - ip'{h)ti2ti']
/
(u) ||Vu|| 4 dfi,
(7.2.50)
>M JM
and more specifically on the coefficient s n-> [(-0'(/i)/i /3 )' - V'(^)^' 2 /i"] (s). By an easy computation, using § W(h)h'3y(s)-iP'(h(s))h'(s)2h"{s)
=
(il/(h(s))h'(s)2yh'(8) 2
= C ($Y(h(s))h'(s)2 = -c2(±y>(h(s)).h>(s)2 Under the asumption that
we obtain that
/
[(i/j'{h)h'3y - rP'(h)h'2h"] («) ||Vu|| 4 d/i > 0 .
(7.2.51)
/M JM
§We have indeed the following computation where we take C = 1, for simplification :
= -Ub)'w»)). Taking the derivative with respect of s of this last equality, we get (^(fc)h«)'(s) = - ( ^ - y ) " (fcW) fe'(«) .
Log-Sobolev inequalities in the strictly convex case
113
Summing up the various contributions, and using in particular the identity ip'(h(s))h'{s)2
+ i/>(h(s))h"(a) = 0 ,
(7.2.52)
which is a consequence of the differentiation of the equality obtained in (7.2.42), we obtain that it is sufficient to prove the inequality XJM iP(h(u(x)))h>(u(x)nVu(x)\f d» < JM ip(h(u(x))) h'{u{x)f (Vu | Hess $ | Vu) dfi 2
(7.2.53)
2
+ ZiifMWMx)))HM*))
l%«(*)| dp .
2
In the case when ip(h)h' = 1 (case when ip = 1) we get the usual Poincare inequality. In the case when ^ — a; In a;, what finally implies the log-Sobolev inequality is the validity of the inequality \!Me-\\Wu{x)\\2dlx < JM eu(Vu | Hess $ Vu) dfj,
(7.2.54)
+ Y,ijIMeU \dau\2dn. In the case of strict convexity, the estimate is quite clear. In the general case, the extension of the log-Sobolev inequality is a consequence of the inequality
*/MWWI V "(*)II 2 <*M ^ IM *(h(l(x))) (Vu(x) I H e s s *(*) I Vu(x)) + E « IM i,(h(l(x))) l% u (*)l 2 dfJ. •
dp
(7.2.55)
Another particular case is the case when $(x) = xp for which the estimate (7.2.55) takes the form A/MKz)l^l|Vu(z)||2dM < IM \U(X)\^{Vu(x) + EijJMK
a;
i
I Hess $(x) | Vu(i)) dp
) l ^ \diju(x)\
2
(7.2.56)
d» .
What we observe in general is that we were not able to use the positivity of the second term in (7.2.55). Remark 7.2.13 As the log-Sobolev inequality, the inequalities (7.2.54) and (7.2.55) have the same nice behavior as observed in the introduction of this chapter. This may be interesting for analyzing the case of M = MN with
114
Log-Sobolev
inequalities
the phase &N\X) = ^j(j>(xj) + $ / when the interaction $ / vex. We have indeed the property that if (7.2.55) is true on dfi = exp —4>(x) dx, then the same inequality is true on IRN measure d^0N) = exp -${0N) efoW, with $(0N)(X) = ^ (xj) . facing the simple question : Under which condition on do we have A JR exp(u(t) - <j>{t)) u'(t)2 dt < fRexp(u(t)-(t))"(t)u'(t)2dt
+ JRexp(u(t)-<j>{t))u"(t)2dt
is conM with with the So we are
. (7.2.57)
The complete answer to this problem seems open outside the strictly convex case. Let us just give here an interesting counterexample.^ Example 7.2.14 Let us consider o an even C°° phase convex at oo but such that o(0) < 0. Then there exists 70 such that Inequality (7.2.57) with = 7^0 is not true when 7 > 70- We can indeed find u such that u = —bjt2/2 in the neighborhood of 0 and u + <j> has a unique non degenerate minimum at the origin giving the contradiction. Here b has to satisfy the condition —^o(O) < b < — 2o(0). The contradiction comes through the use of the Laplace method (with as large parameter 7) which shows that the quantity / exp(u(t) - cj)(t))4>"(t) u'(t)2 dt+ JR
f exp(u(t) - <j>(t)) u"(t)2 dt JR
becomes strictly negative as 7 —*• +00. We note that <^>7 being convex at 00, the corresponding log-Sobolev inequality is satisfied. We shall analyze in Section 7.5 the case of the circle and in the case of 1R we shall give in Section 7.6 some sufficient condition for (7.2.57) corresponding to the non-convex case. ^We thank M. Ledoux for motivating discussions around this problem.
Around Herbst's argument : necessary conditions for log-Sobolev inequalities
7.3
115
Around Herbst's argument : necessary conditions for log-Sobolev inequalities
Herbst's argument is relatively general. We consider a probability measure \i on IR and define for suitable / the expectation E^ by £M(/) =
/
fdp.
JRN
We recall that, for / > 0, the entropy is defined by Ent^f)
= £ M ( / l n / ) - E»(f) In E^f)
.
(7.3.1)
With these notations the log-Sobolev inequalities can be written as Entll(f2)<2CEli(\Vf\2). (7.3.2) We now explain Herbst's argument. We consider a Lipschitzian function F such that ||F||/,i p < 1. (We omit some verification needed for treating general Lipschitz functions and assume for simplification that F is C 1 .) We would like to apply inequality (7.3.2) to f2 = exp(XF) for every X £ M. We first get ^ ( | V / | 2 ) = ^ ^ ( | V F | 2 e x p ( A F ) ) < -E^pXF)
.
Setting H(X) = E^,(expXF), coming back to the definition of the entropy and applying (7.3.2), we get : XH\X)-H(X)\nH(X)
CX2 < -j-H(X)
.
This is a differential inequation that we can transform, by introducing K(X) = i In tf (A), in the form K\X)<^,K(0)
=
Ell(F).
We then immediately obtain by integration that for any X > 0 K{X) = K(0) + / K'(u)du < E^f) Jo Hence, for any A > 0, we get H(X) := ^ ( e x p A F ) < exp (xE^f)
+^ . *
+ ^xA
.
(7.3.3)
Log-Sobolev
116
inequalities
By Chebyshev's inequality, we obtain that, for any A > 0 and any r > 0, KF > E^{F) +r)<
exp f-Xr
+ ~\A
.
(7.3.4)
If we choose the optimal A = -^, in order to minimize the right hand side in (7.3.4), we obtain, for any r > 0 KF > E^F)
+ r) < exp ( - ^
.
(7.3.5)
Observing that we can also play with —F, we finally have the following proposition. Proposition 7.3.1 If the log-Sobolev inequality (7.3.2) for some constant C, then for any Lipshitzian function F, such that \\F\\np < 1, we have f1(\F-Eti(F)\>r)<2exp(^-~y
(7.3.6)
In order to use this estimate (7.3.6), we recall the following lemma. Lemma 7.3.2 Let F be a measurable function on a probability space (X, B, /x) such that for some p > 0, and some constants c,d> 0, we have M
(|F|>r)<2cexp-^,
for every r > 0. Then /
exp a\F\pdn < 1 +
Inird
1 — ad
,
(7.3.7)
for every a < ^. Proof We have indeed, using a simple integration by parts and introducing the function G{r) = f^^dfi, f exp(a\F\P)dfj, = -JexparPdG(r) = 1 + f+°° par?-1 fi(\F\ > r ) e x p a r P dr < 1 + 2ca/ 0 °°(prP- 1 ) exp((a - \)rp)dr = 1 - 2c-^r • a-A
Around Herbst's argument : necessary conditions for log-Sobolev inequalities
117
Remark 7.3.3 (1) We can for example apply this result with F replaced by F — Ep(F) and p = 2. (2) We obtain for example when taking F(X) = \X\, that, if (7.3.2) is satisfied, then
Jexpa\\X\ - E„(\X\)\2d» < [ ± | g ,
(7.3.8)
for any a < ^ . (3) It could be useful to consider the function F(X) — x^. Note that in this case, we only need the log-Sobolev inequality relative to functions depending on one variable Xi. (4) We emphasize that, when considering later problems in large dimension or with parameters, it will be quite important to have all these constants explicit. Exercise 7.3.4 (Counter-example) Take <j>{x) = (1 + | z | 2 ) 5 . Show that for 0 > 1 the Poincare inequality is true and that the log-Sobolev inequality is false for 0 £ [1,2[. Hint Try test functions of the form exp/3, with /3 e]0, | [ . The contradiction is obtained when taking the limit /3 -»• \. More generally, a necessary condition which is obtained in this section (take dfi = exp — <&{x)dx) is the existence of C > 0, D > 0 such that
L
exp —$(a;) dx < C exp -Dr* ,
(7.3.9)
\x\>T
a s r - > +oo. Note that we have, in the case of the counter-example given in Exercise 7.3.4, a strictly positive lower bound for the Witten Laplacian on the 1-forms. But we do not have a uniform lower bound (with respect to J and a) for the family of Witten-Laplacian on 1-forms associated to the family of phases 4>j,a =
+^Jdt2-at.
Log-Sobolev
118
inequalities
In the case when J — 0, one can easily get info- {~^
7.4
+ (*'(*) - «) 2 + ^ " ( * ) ) = 0{a^)
.
(7.3.10)
Extension of the Bakry-Emery argument : convexity at infinity
If there exists a bounded C2 function such that $ + S becomes strictly convex, then one has the log-Sobolev inequality for the measure exp —$ dX. The main trick is given by the identity
IfHm)d"=^lJ[f2lnf-f2lnt-f2+t]d(1-(7A1) It is indeed immediate to see that the function tM.
A/2ln/2-/2lnt-/2 + i ] ^
tends to +oo as t —> 0 and t —¥ +oo and has a unique minimum at t = ||/|| 2 . This permits the comparison of two phases in the same way as for Poincare inequality. We recall that for Poincare we were using var (/)
= / ( / - (f)?dn = \nlRj{f - tfdm .
(7.4.2)
The function t M- / ( / — t)2dfi admits a unique minimum at t = (/). More generally, if we consider for / > 0, the function
m+ 31 H. J [*(/) - *'(*)/ + m'(t) - *(t)] dp, then we get
J (*(/) - * ^ fd^
dp = tinf+ I [*(/) - *'(*)/ + **'(*) - *(*)] ^ ,
(7.4.3) and the minimum is obtained for t = (/). We observe the positivity of (t, y) n- ty(y) - ^(t) - W(t)(y - t)) when * is convex (*"(<) > 0). So we have
Extension
of the Bakry-Emery
argument : convexity at infinity
119
Theorem 7.4.1 Under assumption (7.1.2), if $ is uniformly strictly convex outside a compact, and if ^ € C°([0, +oo[)nC°°(]0,+oo[) satisfies the conditions (7.2.4)(7.2.6), then we have, for some constant C > 0, f * ( / ) d M ~*( JM
f f)
[ * " ( / ) |V/| 2 d/x,
(7.4.4)
JM
for any f > 0 in the class of the Cl functions with bounded derivatives. The proof is indeed based on the following lemma. Lemma 7.4.2 If $ and $ are two phases such that $ - $ = S ,
(7.4.5)
with S bounded, then CL.s(M, exp - $ da) < (exp2 sup \S(x)\) CL.S(M, exp - $ da) . (7.4.6) \ xeM ) This shows that the log-Sobolev inequality is true" for the measure d\i = exp —<& da if and only if it is true dp, = exp —$ da. Proof of the lemma We just observe that, with ^(x) = xln(a;) and for t = (f)fi,
SM*(JW-*{JMf)
(7.4.7)
Exercise 7.4.3 Improve the constant appearing in (7.4.6). Hint Optimize by adding a suitable constant to S.
II In our case, the measure da is the Lebesgue measure dx but the argument is quite general.
120
Log-Sobolev
inequalities
Proof of the theorem We have just to observe that, if $ is strictly convex at +00 on Mm, one can find a C°° function with compact support such that $ + S is convex.
Remark 7.4.4 In the context of the statistical mechanics, one could think that this solves the problem of proving the log-Sobolev inequality for phase of the type :
with (t) := cf)(t) + s(t) becomes strictly convex and with d/iA := exp—$ A we get the existence of a uniformly bounded log-Sobolev constant CL.S.(-^ >djiA) with respect to A. But we get only the estimate CL.s.(MA,dnA)
< fex P 2|A| sup|s(i)A CL.s.(lRA,dflA) V ten J
.
(7.4.8)
This gives the existence of the log-Sobolev inequality but not the existence of a uniform log-Sobolev inequality with respect to A. This property is actually true (for J small enough) but is a consequence of a criterion due to Zegarlinski and of the techniques which will be presented in the two next sections for proving some uniform decay of the correlations.
7.5
The case of the circle
This may be considered as an introduction to what could appear in the case of a compact riemannian manifold. We consider here the simplest case of the circle. The first observation (taking the case of a manifold) is that the log-Sobolev inequality is a consequence of the inequality (expV [|| Hess ^||//. s . + (Ric + Hess <£)(Vi/>, VVO] } > 2e(expV |V^| 2 > , VV-eC^(M;iR). (7.5.1) In the case when M = (M/2Z)N and = 0, the left hand side of (7.5.1) becomes (expV' || Hess V'IIHS ) w m c h dominates ^ j = i ( e x P V ' (aef) 2 ) -
The case of the circle
121
It is consequently enough to treat the case N = 1; hence we have only to analyze the existence of the following inequality /•27T
/ Jo
/»27T
expip(6)\ip"(6)\2d9>2e
expip(9)\ip'(9)\2d9.
(7.5.2)
Jo
It is then natural to make the following change of function h(8) = exp ^2 • We observe that h'(0) = \i>\9)h{9) , and h"{9) = \rl/(9)ih{9)
+
\r{6)h{9)
We have consequently to prove the inequality r2ir
/
•,
/.2TT
(2h"(9) - ^i>'{6)2 h{9))2d0 > 8e /
{h'{9))2d0 .
(7.5.3)
But we have r2lT
=
4th"(9)2d9
+ i/,#)4eKp#)
l
' •j
Then we get, using again the formula for h", C(2h"(9) - \V(9)2 = 4f*"h"(9)2d9
h{9))H9
+ l i c ? V ( * ) 4 expi;(9)d8 -5Jo2V(0)4exPVW0 We then observe that r 27T
= -|/o 2 , r V'W«cpV(0)e».
(7.5.5)
Log-Sobolev
122
inequalities
This gives finally that the log-Sobolev inequality is a consequence of the inequality
{
>Seffh'(0)*dB.
'
We can then use the Poincare inequality on the circle (which is immediate to prove by using the Fourier coefficients of h'), r2ir
\
pin
ti'(8)2de>
Jo
/
h'(e)2de,
Jo
observing that b! is orthogonal to 1. We get (7.5.1) with e = \. Exercise 7.5.1 Determine the values of p €]1,2] for which there exists Ao > 0 such that, for all u € H2^1), we have : A0 / Jo
\u{6)\^\u'(0)\2d6<
\u{6)\^\u"{6)\2d0
/ Jo
.
Solution One first observes that for p = 2, this is simply the usual Poincare inequality. For treating a more general case, one follows almost identically what was done for the proof of the Log-Sobolev inequality. We assume u > 0 for simplification. 2-p
Let us introduce v(9)
=
u{6)^f-i)u'{9)
and we observe that
One can then apply Poincare inequality to v and get
/ v{0)2d8< f
2ir
v'{6)2d6 ,
with v>{8)
= u^(8)
(u"(9) +
J*^u-i(9)u'{0)*y
123
The case of the line
Hence
+ (2^))2/o2^^fT_2W«'W4^ If we observe that
u\efu"{6) = \{u'{.f)'{e), one gets, after an integration by parts in the last term, f*nu&%u"(6)2d0 2
+ {(w%) ~ §^T)((^f) - l))
C^-2uWM
This shows the inequality if
2-* 2(p-l)
\*ZJL-1)<0, 3v(p-l) 5p<6
7.6
The case of the line
Let us come back in the same spirit to the problem of proving (7.5.1) in the case when M = IR. We would like to analyze if for a given <j>, there exists A such that, for any u in C£°(iR), we have
< j R e"W-*(t)^/(t) uHt)2 dt + fR e»(*W(*)«"(t)2 dt.
K
'
Of course, such an estimate is clear (for A > 0 sufficiently small) when 4> is uniformly strictly convex. Let us now assume the strict convexity of 4> at oo, that is the existence of C > 0 and p > 0 such that : 4>"{t) > p > 0 , Vi s. t. |t| > C .
(7.6.2)
Log-Sobolev inequalities
124
When w(t) = u'(t) has its support in {\t\ > C}, it is clear that the inequality is true for X < p. On the other hand, when u> is compactly supported, we have some hope to use the argument given on the circle at least if " is sufficiently small. We shall now try to make this idea rigorous through the introduction of a partition of unity. Let Xi a function in CQ°(1R) such that 0 < *?(*) < 1 , xS(*) = l o n [ - l , + l ] , suppx$C[-2,2].
(7.6.3)
By dilation, we get the existence of Co > 0 and, for any C > O, a function Xi,C with the properties 0 < xi,c(*) < 1 , Xi,c{t) = l<m[-C,+C], suppxi,cC[-2C,2C], C|xi,c(*)l < Co •
(7.6.4)
We now omit the reference to C and introduce X2 > 0 such that Xa+Xi = l -
(7-6.5)
Ix : = J e u ( t ) - * ( t ) x i ( i ) 2 u'(t)2 dt.
(7.6.6)
Let us consider
We first estimate from above 7i by h < Ci(C,4>) [ e"W X l (t)u'{tf dt ,
(7.6.7)
d ^ ^ ^ e x p f
(7-6.8)
with sup
(-^(t))) •
\t€[-2C,2C]
J
What is important is more correctly the variation of <j> because all the argument can be done for cf> replaced by <j>+ const. So we prefer to introduce as new C\ (C, ) the expression d t C ^ e x p f
sup \t,s€[-2C,2C]
(\4>(t) - 4>(s)\)) /
(7.6.9)
125
The case of the line
and observe that the same inequality (7.6.7) is true after possibly a change
of 0. As in the case of the circle, we now introduce
v(t)=exp^-u'{t),
(7.6.10)
and using the standard Poincare estimate we obtain
fxi(t)2v(t)2
< C2(C) [ f Xi(i)V(i)2 dt + f X'i(t)2v(t)2 dt .
J
\_J
J Si
]R.
(7.6.11) Here the behavior of C2 (C) is given by C2(C) = C 2 (l) C 2 .
(7.6.12)
We shall control the second term of the r.h.s. of (7.6.11) later (note simply that the support of x\ is inside the domain of strict convexity of and that C x'\ has be chosen uniformly bounded). Let us analyze Ji:=
[ Xi(t)2v'(t)2dt.
We first observe that v\t) = exp ^ J
(u"(t) + \u'(t)2)
(7.6.13) and consequently
i = fRXi(t)2 expu(t) u"(t)2 dt + i/KXi(i)2expu(i)u'(i)4dt + fRXi(t)2expu(t)u"(t)u>(t)2dt
(7.6.14)
We perform an integration by part in the last term of the r.h.s. (see (7.5.6) in the case of the circle) and we obtain Ji<JBXi(t)2exPu(t)u"(t)2dt -TzfnXiityexvuWu'Wdt - I lRXi(t)x'i(t)expu{t)u'(t)3
(7.6.15) dt.
We now Cauchy-Schwarz the last term and get Ji < fRXi(t)2expu(t)u"(t)2 dt + IIRx'i(t)2expu(t)u'(t)2dt.
(7.6.16)
126
Log-Sobolev
inequalities
We reintroduce exp and we finally obtain the inequality h) [JRexp(u(t)
- <j>{t))u"{tf dt + fRx[(t)2eMu(t)
- *(*))«'(t) 2 dt] , (7.6.17)
where
c3(c,t) = c3Ci(c, yc2,
(7.6.18)
with c 3 independent of <j> and C. We consequently obtain A/i
2
exp(u(t) - 0(t)) u'(*) 2 dt] , (7.6.19)
AC3(C,0<-.
(7.6.20)
X[(t)
which leads first to the condition
This condition is easy to verify by taking, once C is chosen, A small enough. It remains to control /a(A) := A If
(C3(C, )X'i(*)2 + X2W 2 ) exp(u(t) - <j>{t)) «'(*)* * (7.6.21)
which is easily controlled by / 2 (A) <
A
/ (C3{C, <j>)X[(t)2 + X2{t)2W'it)
exp(u(t) - cj>{t)) «'(*) a dt
,
JR
(7.6.22) and obtain /2(A))
<£"(*) exp(u(t) - (j>{t)) u'(t)2 dt
/
(7.6.23)
JR\[-C,C]
where C4(C,p) = C4™*i£i<£>*£Jl. We get the new condition that Ac4max(C1(C^)2)l)<|.
(7.6.24)
For any choice of <j> and C such that (7.6.1) is satisfied for some p > 0, we can find A satisfying the two conditions (7.6.20) and (7.6.24). We now
127
The case of the line
introduce !,(# = s u p ( - 0 " ( t ) ) + , ten
(7.6.25)
and we obtain Xjeu(t)-Ht)\u'(t)\2dt
< i / e «(*)-^(t)0"(i) |u'(t)| a cft + 1 J e»(*)-*W | u " ( i ) | 2 dt +7? A C 4 (C, p) / <•«(*>-*(*) | u ' ( i ) | 2 di .
(7.6.26)
We have obtained the proposition : Proposition 7.6.1 Let us assume that <j> satisfies (7.6.2). Then, for a given universal constant Cs, the inequality (7.6.1) is true for A small enough if there exists C such that c 5 (max(C 1 (C, 0), l ) ) 2 r,{) < ^ ^ -
,
(7.6.27)
where p(C,(f>) = inf{ t | \t\>c} "{t) > 0- As a consequence the log-Sobolev inequality for the measure exp —<j>{t) dt. Although this result is not empty and corresponds to a natural notion of weak non-convexity, the condition (7.6.27) is technical and in particular not adapted to a proof of the log-Sobolev inequality in large dimension. Note also that the argument given in Section 7.4 gives the log-Sobolev inequality under the assumption (7.6.2). Example 7.6.2 For tj>(t) = 7(t 2 — /3)2 + at and fixed (7, a), there exists C (in the form C6\/j3) such that the condition (7.6.27) is satisfied if fi > 0 is small enough (and uniformly for a in bounded interval). Remark 7.6.3 Various other ideas can be also tried. For example, one can look for a change of variable such that the phase after the change of variables becomes strictly convex. This will be proposed as an exercise in the next section. But this does not permit to treat the previous example uniformly with respect to a : this seems an interesting open problem.
128
7.7
Log-Sobolev
inequalities
General remarks
A general criterion for proving log-Sobolev inequalities was given by B. Zegarlinski and based on the proof of uniform decay estimates for the pair correlations. We shall analyze these estimates in chapter 8 and give the proof of the criterion in chapter 9. Another interesting property (that we leave as an exercise in this course) is Exercise 7.7.1 If we denote by XLS the best A such that the log-Sobolev inequality (7.2.2) is satisfied then show that ALS < A2 - Ai , where \j corresponds to the j'-th eigenvalue of A$
(7.7.1) (or A$ ).
Hint Apply the log-Sobolev inequality with u = 1 + ef and then consider the limit e —> 0. On the other hand we have shown that the lowest eigenvalue of the Witten Laplacian Xwi on 1-form, satisfies also \wi < A2 - Ai .
(7.7.2)
In the uniformly strictly convex case we have seen that, with A0 = inf A m i n ( Hess $(a;))) , X
\LS > Ao ,
(7.7.3)
and that, the lowest eigenvalue of the Witten Laplacian on 1-form, satisfies also XWi > Ao •
(7.7.4)
But the counter-example 7.3.4 has given a case when XLS — 0 and Xw% > 0. Exercise 7.7.2 Let us consider a global diffeomorphism x H-> y{x) from M on M. In order to analyze the log-Sobolev inequality relative to the measure on M d/j, := exp —<J)(x)dx, one would like to use the change of variables associated with this diffeomorphism and apply the theorem obtained in the strictly
General
129
remarks
convex case. Show that this strategy can be efficient for v(x) = x4 + vx2 with v < 0, \i/\ small enough. As an intermediate step, find a differential inequation involving (/> and a weight b such that, for some A > 0, we get the following inequality : \(
fu2\nu2dn-
((
u2d^\\a((
u2dp\\
< f b{x)2
\u'{x)\2dfi.
In a second step, one can analyze the case when <j> = <$>v and b(x) = , In a last step, show that one can improve the result by considering the
weight h{x)
•= Tnfe-
Solution As suggested in the exercise, one can perform the change of variable x = 6(y) (whose inverse is denoted by i ^ y(%)) and we consider v(y) = u(9(y)). One can apply to v Log-Sobolev inequality relative to the measure exp —i/j(y)dy ^(y) =
cf>(9(y))-ln(0'(y)).
If ip is strictly convex,
r(y) > P , we can apply the Bakry-Emery argument. The right hand side of the log-Sobolev inequality becomes, when coming back to the initial coordinates, p ( / u2 In u2dfx - (J u2dfi) ln(J u2d/j,)) < je'(y(x))2u'(x)2exp-(i>ix) dx . One consequently obtains the intermediate estimate by considering b(x) = 6'(y(x)), supposing in addition that 6' > 0. Let us now make explicit the various conditions. One would like also to have b(x) < C in order to come back to the standard log-Sobolev inequality. The condition of strict convexity becomes
By reintroducing b, we get (sc)'
F(x) := 4>"(x)b(x)2 + <}>'(x)b{x)b'(x) - b(x)b"(x) > p .
2.
Log-Sobolev
130
inequalities
Let us now consider the specific example b(x) = ( 1 + 1 x a) ; , which corresponds to the change of variables y'{x) = ^/(l + x2). We note that y(x) (which can be normalized by y(0) = 0 and is then an odd function) behaves as \x\ is large like ±x2. The condition (sc)' is satisfied as x —> +00; we have indeed lim
F(x) = 8 .
|x|-y+oo
F being continuous, it suffices to determine for which i/'s the function F(x) is everywhere positive. It is easier to consider (for coming back to the analysis of the sign of a polynomial) to G(x) := F(x)b(x)~6. An easy computation gives : G[x) = (12a;2 + 2i/)(l + x2)2 - 2(2a;2 + v)x2{l + x2) + (1 - 2x2) . Introducing z = x2, we get H(z) := G(x(z)) = 2(6z + i/)(l + z)2 - 2z(2z + i/)(l + z) + (1 - 2z) = 8z3 + 20z2 + (10 + 2v)z + (1 + 2v) . We now observe that H(0) = {\-\-2v), ff'(O) = (10+2v), H ' ( - l ) = - 6 + 2 i / et lim|2|^ + 0 0 H'(z) = +00. This implies that H{z) > 0 if v e] - 5,0]. It is rather easy to show that considering 6^(0;) = (1 + 01'x2)~ 2, leads, for a suitable (3 > 0, to an improved condition on v for getting the logSobolev inequality. Comment Of course, we know by other means (using the convexity of v at 00) that the log-Sobolev inequality is valid for any v. So this remains an exercise with no actual application for the moment.
7.8
Notes
Section 7.1 • In this chapter we were essentially inspired by three references : the book by J.-D. Deuschel and D.W. Strook [DeStl], the notes of a course by M. Ledoux [Ledl], and some unpublished preprint by A.Val. and A.Vict. Antoniouk [AAl].
Notes
131
• Other interesting references are the course by G. Royer [RoyG2] (in french) and the collective book [Aandall]. Section 7.2 • The main result is due to Bakry-Emery [BaEm]. • Of course, condition (7.2.1) is the condition appearing in the flat case :M — IRN. In the case when M is a compact riemannian manifold, we have to add the Ricci curvature (cf [AAl], [AA2] and [DeStl]). • We follow here some variant of Bakry-Emery result due to [AAl]. The advantage is that we prove in the same way Poincare's inequality and Log-Sobolev inequality. • We have chosen to concentrate our study on the case when M = MN. The case when M is a compact Riemannian manifold is treated in [DeStl]. • For Remark 7.2.5 we refer to [GUa] who refer to the FeynmannKac formula, which is the probabilistic version of the Trotter-Kato formula and to the discussion in Reed-Simon [RS-IV] (p. 201-208). • For the local regularity for elliptic operators, we refer to any book in PDE and in particular [LiMag]. • For the proof of Theorem 7.2.9 and the theory of globally hypoelliptic operators, we refer to [Hel] and references therein. See also in the context of Witten Laplacians [Jo]. Section 7.3 • Herbst's argument is the content of a letter of I. Herbst to L. Gross as initially transmitted by Gross. • Here we follow the nice exposition by M. Ledoux [Ledl]. We have just made more explicit some constants appearing in the arguments. • Lemma 7.3.2 can be found in Ledoux [Ledl] (Proposition 1.2). • The hint in Exercise 7.3.4 is proposed in Deuschel-Strook [DeStl] (p. 269) (see also [Ledl]). Section 7.4 • This idea is mentioned** p. 267-268 (See Corollary 6.2.45) in Deuschel-Strook [DeStl]. The main trick is given p. 248 in [DeStl]. **We thank T. Bodineau for this remark.
132
Log-Sobolev
inequalities
Section 7.5 • We follow here an elegant proof due to Emery-Yukich as presented in Deuschel-Strook [DeStl]. Section 7.6 • Remark 7.6.3 was analyzed in some unpublished paper of Helffer [Hel7] as a variant of similar ideas presented in [AAl]. • Other results based on the use of the Hardy inequality have been obtained by I. Gentil and C. Roberto [GeRo] (see also [Aandall] and [Ge] (Proposition 2.3.7), which is connected with [BobGo]). Section 7.7 • The argument in Exercise 7.7.1 can be found for example in the course by Ledoux [Ledl]. • Exercise 7.7.2 and the comments are also inspired by discussions with V. & V. Antoniouk around [AAl].
Chapter 8
Uniform decay of correlations
8.1
Introduction
As presented in the general introduction (Section 1.2), our aim is to analyze the thermodynamic properties (and particularly the decay of correlations) of measures coming from statistical mechanics. Here we more precisely consider the case with boundary, that is measures of the form exp—$A'W (X) dX in the case when $A>W, which is associated with cubes K
*A'w(X) = 5>fo) + f
£
jeA
({j}u{fc})nA#0 , j~k
\zj-ZkW
(8.1.1)
where • X = (xj)jeA, • is a one-particle phase on M with at least quadratic increase* at CO,
z
3 =XJ Zj = WJ
if
J £A, if j £ A .
/
8 1 2
x '
"One can in some case consider a weaker assumption but we always assume that J exp —4>{t) dt < +oo. In any case, this condition on is not far to be necessary for the log-Sobolev inequality as discussed in Section 7.3. 133
134
Uniform decay of correlations
• j ~ fc means that j and fc are nearest neighbors t for the i1- distance in^d. We shall sometimes use the following decomposition $A.<" = $ A
$ A,<- ;
^8>1_3j
<#(X) = £ > ( * , - ) ,
(8.1.4)
+
j
with J'€A
and $A'"(X) =
2 \zj-zk\2. ({j}u{fe})nA,40 , j~fc
(8.1.5)
We could also consider the free boundary condition which was considreed in the introduction and corresponds to the phase $ A . / = $ A + J $A,/
(816)
;
with *ff(X)=
£ j,k£A
K-^l
2
-
(8-1-7)
, j~k
If necessary, the dependence on J will be mentioned by the notation <J>A,w _ $A,w„7
Qr
<j>A,/ _
$AJ,J
Let us now state the assumptions on the single-spin phase . We assume that 4> is C 2 on 2R and convex at oo, so there exists C > 0 such that "{x) >^,\/xeM
s.t. |a;| > C .
(8.1.8)
We assume also the technical condition that <> / is C°°, and that there exists p > 0 and, for all fc € W, C*, such that, |^ (fc+1) (z)l < Cfc < (1""fc)+ ,Vx€lR,
(8.1.9)
where, for u G 2R, < u >:= (1 + \u\2)i and, for t e JR, (t)+ :— max(i, 0). tOne can also analyze the case when j and fc were nearest neighbors in A considered as a discrete torus.
135
Introduction
The typical example will be {x) = ±\x4
+ l-ux2 .
(8.1.10)
where the parameters A and u satisfy A>0.
(8.1.11)
We would like to analyze the possibility of having any sign for v. The sum in (8.1.7) is over the non-oriented pairs of A x A. Our main problem will be to analyze the properties of the measure d/iA,a, := e x p - ^ ' ^ X ) dX/ [ / \J(R»)
e x p - $ A ' " ( X ) dX ) ,
(8.1.12)
or of the measure dfiA := exp -$AJ(X)
dX/ [ [ exp -$A'f(X) N \J(R )A
dX ) . J
(8.1.13)
We shall in particular analyze the covariance associating to f,g£C%mp(lRA) CovAiU,(/,<7) = ( ( / - (f)A,u)(g - <<7)A,W))A,U,
(8.1.14)
where ( • )A,W denotes the mean value with respect to the measure dfi^^ and C£mp(]RA) is the space of C°° functions with polynomial growth. Similarly, we associate with $ A '^ and the measure d(iA the mean value (• )A and the covariance COVA. Our main theorem is the following : Theorem 8.1.1 Let us consider, for any UJ E JRZ and A C 2Ld, the phase $ = $ A >"" 7 = $ A + J$i 'u with cf> satisfying (8.1.8) and (8.1.9). Then there exists a constant C and JQ > 0 such that, for any J G [0, Jo], for any u>, for any A c F and any tempered functions f and g on M , we have : I CovA,M,9)\
< C ( e x p - i d ( S A , S A ) ) \\dAf\\L2 \\dAg\\L2 .
(8.1.15)
In this case, we say shortly that we have uniform decay estimates. Here SA (which is called the lattice support of / in A) is the smallest subset of A such that f(X) = f(Xsf) where / is a function on Msf . For example the support of f(X) = Xi is {i}.
136
Uniform decay of correlations
Remark 8.1.2 In particular, when / = g, we recover Poincare inequality. Considering the special case when f = Xi and g = Xj, we get for the pair correlation introduced in (1.2.6) : Corollary 8.1.3 Under the same assumptions as in Theorem 8.1.1, there exists Jo > 0, C and K, such that, for any A, u> and J £ [0,+Jo], the correlation pair function satisfies I CorAtU(i,j)\ 8.2
< Cexp-Kd(i,j)
, Vt.j £ A .
(8.1.16)
Lower bound for the spectrum of the W i t t e n Laplacian
Let us recall that the Witten Laplacian on 1-forms attached to the phase $ = ^A'u is defined as, A<1} := * -V
(
^J'€A ^
9
i ' ^ U r
9
dxj ~ 2 dxj ) \ dxj
i i a$ ~\
-
<~ 2 dxj J
)I + H e s s $ .
V(8.2.1) ;
2
defined on the L 1—forms with respect to the standard Lebesgue measure on Mm, with m = |A|. As already said, this Witten Laplacian A$ is essentially selfadjoint from CQ° under the assumption that $ is C 2 and there is consequently a uniquely well defined selfadjoint extension which coincides in particular with the Friedrichs extension. The aim of this Section is the proof of the following theorem : Theorem 8.2.1 For any A c Zd and w £ M**, let $ := $A satisfying (8.1.8)-(8.1.9). There exists JQ and a\ > 0, such that the lowest eigenvalue A 1 ''" ; ', of the corresponding Witten Laplacian on 1-forms A$ , satisfies, for any cube A, u £ M and J £ [0, Jo], A A '"' J >
(8.2.2)
Similarly, we have also : Theorem 8.2.2 For any A C 2Zd, let $ := $A satisfying (8.1.8)-(8.1.9). There exists Jo ando\ > 0, such that the lowest eigenvalue
Lower bound for the spectrum of the Witten
137
Laplacian
Ai , of the corresponding Witten Laplacian on 1-forms A$ , satisfies, for any cube A and J e [0, Jo], \*'f'J
> ax .
(8.2.3)
The starting point for the proof is the basic identity ( A « u | u)L, = £
\\Xku^
+ W
^ L u , .
Uk
dX ,
(8.2.4)
with Xj = dJ + \di$
(8-2-5)
.
and dj
=
d
"
=
dx~
(8
•
-2'6)
Let us denote by w]j ' and w^ ' the single-spin Witten Laplacians (respectively on 0- and 1- forms) attached to the variable Xj and the phase on M Mxj)
:= &*>>*(X) .
(8.2.7)
The function 'j depends actually only on the ze with £ ~ j and let us recall that zi = xe if £ e A and zi = cj( if £ e ZSd \ A. We have indeed
4>i{t) = <j>{t) + J
J2
\t-ze\2
+ $(Zj)
(8.2.8)
({<}u{j})nA^0 , e~j
where the last term is independent of t and will be irrelevant in the discussion. The operators u>\ ' and «;] ^ depend only on the zi with £ ~ j . It is quite important that the estimates are proved independently of these parameters. We note the relations (o)
w3
: X* Xj ,
(8.2.9)
and
wf)=xjx; = x;xj
+0.
(8.2.10)
138
Uniform decay of correlations
According to the context, we shall see these identities as identities between differential operators on L2(IRA) or on L2{MXi) (the other variables being considered as parameters). With these conventions, we have
I «)L2 = Ejtk€A , j ^ f c r f ui I vi) + +J {Kess'$iU \ v) ,
EJ€A(WJ1)UJ
I vj)
K
o in ' ' '
(8
where $* = $ i 'u denotes the interaction phase and Hess ' means that we consider only the terms outside of the diagonal of the Hessian, that is such that for k, £ £ A ( Hess '*<)« = - 1 ifk~e = 0 else .
\ • • )
Here we observe that Hess $j is independent of z and that J Hess 3>j corresponds to a perturbation in 0(J~), where O is uniform with respect to A, using the discrete Schur's Lemma. Note that we get from (8.2.12) the inequality
(A^u | u)L2 > J2(wj1)ui I ui) + J ( H e s s ' $ iU | u) .
(8.2.13)
J'6A
We get consequently the following theorem : Theorem 8.2.3 Let us assume that there exists pi > 0 such that, for any j £ Z z S M Wt t the operator* uA ' satisfies W(P > Pi ,
and any
(8.2.14)
then, for any e > 0, there exists Jo > 0 such that the Witten Laplacian Wf, with $ := $A.<"»7' or with $ := $ A - / ^ , satisfies for any A C Zd, u)£Mzd andje [0,+J0], A ^ > (pi - e) .
(8.2.15)
We recall that the positivity of ur- ' is immediate from the definition. We also observe that it is sufficient to treat the case of a fixed jo all the families being unitary equivalent (after a simple change of the names of the parameters). *For a
given j £ %**, the effective parameters are actually the %i such that t ~ j .
Uniform estimates for a family of 1-dimensional
Witten Laplacians
139
The other important point is to verify the condition of uniformity. This will be done in Section 8.3. 8.3
Uniform estimates for a family of 1-dimensional W i t t e n Laplacians
We shall discuss various conditions under which these uniform estimates can be obtained. In all this section is at least C2. If ip is a C 2 phase, we shall denote by wA' and uA the corresponding Witten Laplacians defined in this simple case by (0)
i«') 2 -^"< ( )-
(8.3.1)
-£+i*w+!#•<«>.
(8.3.2)
wy = - ^
+
and
These two operators being positive are automatically selfadjoint on L2(M) starting from CQ° as recalled in Chapter 6 (Theorem 6.6.2). The first condition is (for reference to the strictly convex situation which was analyzed through the Bakry-Emery argument) the existence of C > 0 such that (sc)
"(t) > ^ , Vi G M .
(8.3.3)
A weaker condition is (sc(oo))
"(t) > ^ , Vt € 2R , |t| > C .
A still weaker assumption is that there exists a bounded function x such that (scm)
(j>"{t) + X"(t) >^,yteIR.
(8.3.4) m
C2
(8.3.5)
These three conditions are ordered : (sc) => (sc) (oo) => (scm) .
Another family of conditions corresponds to assumptions on the operator
140
Uniform decay of correlations
or more precisely to the quadratic form associated to wred on C£° : u H> qTed{u) = (wredu | u) .
(8.3.7)
We consider a new family of conditions starting with : (qsc)
qred(u) > 1 ||u|| 2 , Vu G C%°(1R) .
(8.3.8)
o A weaker condition is (qsc(oo))
^ | | u | | 2 , Vu G C0°°(iR\] - C, +C[) .
(8.3.9)
Finally a still weaker assumption is the existence of a bounded function x in C 2 such that (qscm)
q™d(u) + \ J
X "(«)
\u(t)\2 dt>~
\\u\\2 , Vu G C0°°(2R) . (8.3.10)
These three new conditions are again ordered : (qsc) =$> (gsc(oo)) => (qscm) and it is also clear that (qsc) is weaker than (sc). The condition (qsc) permits to treat roughly speaking functions which are strictly convex in mean value. Explicit criteria exist for verifying (qsc) and are related to sharp versions of the Garding inequality. Remark 8.3.1 It is important to observe that if satisfies one of the six conditions (sc), ...., (qscm(oo)), then the same condition will still be true for the phase <j)jta(t):=4>(t) + Jdt2-at
(8.3.11)
for any J > O. More generally this is still true for any J such that 2Jd>-—.
(8.3.12)
o # In the preceding section, we have shown that the proof of a uniform lower bound for the Witten Laplacian A*/ can be deduced from the study of
Uniform estimates for a family of 1-dimensional
Witten Laplacians
141
one-dimensional Witten Laplacians. We want to analyze W
*L := -§> + i(^."(*))2 + \WjJt)) ,
(8-3.13)
with a = -2j^2zk .
(8.3.14)
o~fc The first theorem is the following : Theorem 8.3.2 Let be a phase satisfying (8.3.8). Then, there exists Jo, and pi > 0 such that, for any (a, J) 6 2R x [0, +Jo], we have w$la
>& .
(8.3.15)
The proof is trivial if we observe that w{£a>wred
+ Jd,
(8.3.16)
in the sense that we compare the corresponding quadratic forms on CQ°. Note that in the strictly convex case (sc), we simply write w(£
a
><j>" +2 Jd.
(8.3.17)
The second theorem is : Theorem 8.3.3 Let 4> be a phase satisfying (8.3.10). Then, there exists Jo, and pi > 0 such that, for any (a, J) £ M x [0, +Jo], we have «#].„ > Pi •
(8-3-18)
We have the following lemma (cf what we did in Chapter 7, Lemma 7.4.2). Lemma 8.3.4 If, for a phase ip in C2, Xi(ifj) is the bottom of the spectrum of the Witten Laplacian i/A attached to rp, then, for any C2 functions \ and such that X is bounded, we have \i(4)
> (exp-2||x||z,-) • Ai(^ + x) •
(8-3.19)
Uniform decay of correlations
142
The proof consists just in observing the identity (w^u\u)
= (exp - f ( - £ + i ( ^ + x')) exp § u | e x P - | ( - S + 5(^ + x'))exp|u),
with ip = ^j-.aWe then deduce, that, for any u 6 (w^ulu)
, ^
, • ^
CQ°(1R),
> exp - | | X | | L ~ (W£+X exp f u| exp f u) > e x p - | | x | U ~ Ai(^ + x) || exp fu|| 2 ,
,g 3
^
and consequently (w^u\u)
> e x p - 2 | | x | U ~ A i ^ + x)!!"!! 2 •
(8.3.22)
The minimax principle and (8.3.22) give (8.3.19). This ends the proof of Lemma 8.3.4. Remark 8.3.5 A second proof consists in working with the second eigenvalue A2 of i/A . For the proof of Theorem 8.3.3, we can then use the lemma and apply the previous theorem with <j>a,j replaced by (j>a,j + X8.4
A proof of the decay of correlations
We now present the proof of Theorem 8.1.1 on the decay of correlations and show that one has just to use the uniform Poincare inequality for the Witten Laplacian associated to the single-spin phase uA ' (in other words the existence of a uniform lower bound for the second eigenvalue A2 (a, J) of this Laplacian) instead of the uniform control of the lowest eigenvalue X{^(a,J) of the singl e-spin Laplacian w^ ^ on all 1-forms. This makes no difference in our case because our single particle phase are denned on M but we take a point of view which is useful for future extensions to the case when <j> is denned on M . Let us denote by L\ the Hilbert space L%:=L2{lRA;ex-p-$dX) and by £1$ the Hilbert space of the A:-forms with coefficients in L\. We were starting from the operator A§ = d*'®d where d is the differential
A proof of the decay of correlations
143
on the 0-forms and d*'* is the adjoint defined on the one-forms : (du | c) n i,2 = {u | d*' c) L 2 , for all u G C^°(MA) and
/-(/)=4V
(8.4.1)
from which we get the identity
df = d 4>0) u = A^ du .
(8.4.2)
We have in mind to take f = Xi, g = Xj with i and j in A. The idea consists in the introduction of weighted spaces on A, associated with strictly positive weights satisfying exp -K < p(£)/p(k) < exp K ,
(8.4.3)
where £ ~ k (this means that £ and k are nearest neighbors in Zd) and K will be determined later. For a given i € A, the function p(£) = exp—K,d(i,£) where d is a usual distance on M satisfies this condition. For a given A c 2 with |A| = m, let us now associate with a given weight p on A the m x m diagonal matrix M — M A denned by MM = 6M p{£) , for £, k e A .
(8.4.4)
Note that with this choice, we get, for f = Xi and g = Xj, (Mdg)t = Sji exp -Kd(i,j)
, {M~ldf)i
= Su ,
for our specific choice of / and g. We consequently can write Cov A ,w(/>s) = (du-dg) = {{M-ldu)
• (Mdg))
(8.4.5)
We have consequently to control cr := M~xdu .
(8.4.6)
Uniform decay of correlations
144
In order to do that, we rewrite (8.4.2) in the form M-^df
= M-XAYMM-Xdu = Axa + {M~lAxM - Ai)
^' " '
We now take the scalar product in Cl$ with a in the identity (8.4.7) and get ((M-'df)
• a) > {(A.a) • a) - CJ\\a\\2Ql,2
.
(8.4.8)
Here we have used the pointwise estimate of | | M - 1 Hess 3>jM|| in C(£2(A; M)) . We observe indeed that, for all X £ MA, || Hess $ ( * ) - M - 1 Hess *(X)M\\C{P) = \J\ || Hess $i(X) - M'1 Hess ^i{X)M\\c{p)
(R,q,
.
K
'
In this example, observing that the coefficients of 5M( Hess $j) =: Hess $ j - M - 1 Hess $*M vanish if k ^ £, it is immediate, using Schur's Lemma, to get that \\SM( Hess $i)\\C(P) < 2d sup^ f e |(1 - PW f$)\ < 2d max ((1 - exp - K ) , (exp K - 1)) = 2d6 ,
(8.4.10)
with, 0:=(expK-l),
(8.4.11)
and this is clearly uniform with respect to the lattice. Now we can rewrite (8.4.8) in the form ((M-'df)
• a) > £ < ( < # > , ) • aj) - C7||<7||» i>a , i
(8.4.12)
where a\ ' is unitary equivalent to Wj ' (through the map u >-¥ exp — ^ u>) and C is uniform with respect to all the parameters. The crucial observation is that
= dj(p{j)-iu)
145
A proof of the decay of correlations
is consequently exact§ as a function of Xj in JR. Using this property, we can write {{M-ldf)-a) > EpOr 2 H«?>lla^ -CJ\\o\\lY
,
(8.4.13)
j
where C is uniform with respect to all parameters and a£> = <£•*' dj = (-dXj + dXj<j>j)dXj
(8.4.14)
which is unitary equivalent (through the conjugation u i-> exp — ^ u) to Wj ' and will be more shortly denoted as a^ '. We would like to consider (in addition to the parameter w) the Xk (k ^ j) as parameters (we write Xj) and consider the single spin Hilbert spaces of fc-forms on M constructed
onLl{M):D.k^. We recall the observation that {dXi<j>j){xj) = (dXj)(xj,Xj)
.
In particular, the operator aj ' initially defined as a selfadjoint operator on L\{M ) can also be considered as a family (depending on the parameter Xj) of selfadjoint operators on L?.(iR). We denote shortly by Uj the map Xj t-¥ U(XJ,XJ) and define Uj = Uj — (UJ)J where the mean value ( • )j is respect with the measure on JR exp —j dxj. We have of course djUj
— djUj ,
and aKj 'UJ
(0)~
= a} Uj .
Using the standard inequalities, we |get, denoting by Aj the second (or the first non zero) eigenvalue of a;- ' (or equivalently of vn ')1 considered as an operator on L2(M) \ ( 0 ) ; j , z f c | | i|2 _ \(o);j,z f c 2 | | < 7 j | | n i , 2 — <*2 ^3
A
{a{pUj | Hj)L2
< A < 0 ) ; ^ II a
( 0 ) - II
\\ ) 2
< \Wfuj II
§ This is of course automatic in dimension 1.
U
J\\LI
I I - II
•IIWJIILJ
(8.4.15)
146
Uniform decay of correlations
We then multiply by exp — ( — j) this inequality, integrate over (JR)A\{J'I with respect to the other variables Xj, and obtain
((M-Mf)-^) > £ ( mf'A 0)ii, " k ) IWiWl^-CJWaW^,,.
(8.4.16)
For a suitable constant C and for J > 0 small enough, we get, after use of Cauchy-Schwarz in (8.4.16), the inequality
IKM-^JUn^ • |k|| ni . a > ^ I M I ^ ,
(8.4.17)
HM-^ulU, < CHM-^Hn^ •
(8.4.18)
and finally
The end of the proof is the consequence of I CovAjU(f,9)\
< IIAf-^uHnw • ||Afdff||ni.a ,
(8.4.19)
and (8.4.18). We choose now the weight p = expKd(S$,-)
,
(8.4.20)
for implementation in the matrix M. This ends the proof of the theorem. Remark 8.4.1 In the next chapter, we shall meet a slightly more general situation. The considered functions / and g will a priori be defined on M instead of M . Their supports Sf and Sg will always be assumed to be finite. In this case we apply (8.1.15) to the functions fu (and g^) defined by fu(XA)
= f(X)
, with X = (XA,XA°)
, XA° = uA') .
We observe that in this case
In order to avoid heavy notations we shall continue to write (8.1.15) in the same way.
Generalized Brascamp-Lieb
inequality
147
Exercise 8.4.2 For the model 4>{x) — xi + vx2, discuss in function of v, i, j , k,l the quantity (xiXjXkXt)A,j — {xixj)A,j{%kXi)A,J, uniformly with respect to A C Z , where {-)A,J corresponds to the mean value with respect to the measure exp— $\(X)dxA where
• $A(X) = E P 6 A 4>(xp) +JE{p~q,p,geA}
K ~ x<,\2>
• J G [0, Jo] with Jo small enough (independently of A). Remark 8.4.3 When d = 1, one could expect from the transfer matrix approach that we have exponential decay. What seems more difficult is perhaps to get by this approach the uniformity with respect to UJ. The preceding remark leads to : Exercise 8.4.4 Relate the splitting for the transfer operator and the lowest eigenvalue of the Witten Laplacian. Remark 8.4.5 We do not consider in this course the problem of analyzing the existence and the uniqueness of a limit measure as A —» 2Z . This leads in particular to the notion of Gibbs measure.
8.5
Generalized Brascamp-Lieb inequality
In this section we shall show how to improve the decay estimates by taking account of the possible growth of <j>" at oo. This could be quite useful for considering more general interactions. Our aim is to show the following theorem : Theorem 8.5.1 Under the assumptions of Corollary 8.1.3, there exists C > 0, J0 > 0 such that, for any A c Z , any J G [0)k7o]; o,ny w G M , and any tempered functions f and g on MA,
I CovAi„(f,g)\
||e.d A /|| 19 ||e-d Aff || La , (8.5.1)
148
Uniform decay of correlations
with (Q(X))jk
= {"{xj) + C)~Ujk
.
(8.5.2)
Proof Of course C is chosen such that the following condition <j>"{t) +C>l,\/teM.
(8.5.3)
Let us introduce the matrix Q(X) defined by (OA(X)) = 0"1 .
(8.5.4)
In analogy with (8.4.18), we shall show the inequality ||0 o M ^ d u l L i ^ < C||0 o M
- 1
#|LM •
(8.5.5)
I Cov A , u (/,fl)| < lieAf-^uHni^ • ||eAfdff||ni.2 .
(8.5.6)
The end of the proof is the consequence of
The proof is then easy, following the lines of our preceding proof and using in addition (8.3.17). We have just to improve (8.4.16) in ((M-'df)
• a) > i | | e
.
(8.5.7)
This implies, using Cauchy-Schwarz (8.5.5).
8.6
Notes
Section 8.1 • Most of the material presented in this chapter is taken from the series of papers [Hel9]-[He2l] (See also references therein), which were some elaboration of [HeSj5] to a non-convex situation. In the same spirit, let us also mention the paper by [BaJeSj]. • The analysis of the decay of correlations has a long story and we mention in particular [Do], [Gro2], [Fo], [Ku], [Sok] and [De].
Notes
149
Section 8.2 • Theorem 8.1.1 is proved in [He2l]. Without the uniform control with respect to w, it was also obtained by Bach-Jecko-Sjostrand [BaJeSj]. Further much finer results on the correlations are given in [Sj7], [Sj8] and [Sj9]. We were influenced in the final version of the proof by discussions with J. -D. Deuschel. • Although the Witten Laplacian point of view seems to us interesting (and this is J. Sjostrand who was the first to emphasize its role, which was already present in [HeSj5], in [Sj8]), one should not overestimate the advantage of this approach in comparison with the Dirichlet Laplacian point of view and with what was done in Geometry with the Bochner Laplacian and its role for getting the Lichnerowicz formula. • The strict positivity of the Witten Laplacian on 1-forms is a consequence of general arguments (see J. Sjostrand [Sj8] and J. Johnsen [Jo]. But this proof does not give automatically the uniformity of the lower bound with respect to parameters. Section 8.4 • The uniform decay when d = 1 has been proved by B. Zegarlinski [Ze]. For the notion of Gibbs measure we refer to the work by Dobrushin and collaborators (see [BeHo] and references therein, [AA2] and [De]). Section 8.5 • This extension is taken from a common work with T. Bodineau [BodHel] and partially inspired by [AA2]. Similar estimates occur in an unpublished paper by Bach-Jecko-Sjostrand. • Other extensions are due to Bach-Moeller (in preparation).
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Chapter 9
Uniform log-Sobolev inequalities
9.1
Introduction and preliminaries
We recall that our aim is to analyze the thermodynamic properties of measures exp—$ A ' W (X) dX in the case when $ A , a j , which is associated with subsets A C 2Zd and some u G Mz defining the boundary condition, which was introduced in (8.1.1). Our main assumption is an assumption of convexity at oo of the single spin phase . We assume that there exists a bounded C°° function s such that <j> := 4> + s is strictly convex, that is that there exists p > 0 such that ( + s)"(t)>p>Q,VtelR.
(9.1.1)
Our main problem will be to analyze the properties of the renormalized measure dEA'" := Z~K]U • exp - $ A - " ( X ) dX ,
(9.1.2)
with ZK,U:=
(
exp-^u(X)dX.
(9.1.3)
More specifically we would like to analyze under which conditions we can prove the existence of an uniform log-Sobolev inequality attached to this family of probability measures. For this, we have first analyzed in the previous section the covariance associating to / and g e C£mp(IRA) : EA'"(f;g) := Gov A , w (/,
152
Uniform log-Sobolev
inequalities
Our aim is to prove the following theorem mainly due to Zegarlinski with additional arguments of N. Yoshida and Bodineau-Helffer. T h e o r e m 9.1.1 Let us assume that (9.1.1) is satisfied. Then there exists a constant c in ]0, +oo[ such that for any A C Zd and any u> £ JR. , we have (/ln/) A ,u, < 2c(|V/5| 2 ) A , w + (f)AtU
\n(f)AtU1 ,
(9.1.4)
for all nonnegative functions f for which the right-hand side is finite. R e m a r k 9.1.2 We will use in the proof that in Theorem 8.1.1, we can consider more general cylindrical functions (that is with finite lattice-support) on IR . I n this case (See Remark 8.4.1), one has to consider that Sf:=Sfr\A,
(9.1.5)
and V / has to be replaced by V A / where V A / := (<9Xj/)ieA- Then inequality (8.1.15) becomes an inequality between functions over MA", the variable in M being denoted by LJ (actually uA"). It is in general difficult to keep all this interpretation in the notations.
9.2
Some log-Sobolev inequality for effective single spin phase
One basic step in the proof of uniform log-Sobolev inequalities is some proof of log-Sobolev inequalities with a weaker control with respect to the size of the "support" of the function. L e m m a 9.2.1 Let us assume that satisfies (9.1.1), then there exists J§ such that, for any n £ IN, there exists a constant* Cn such that, for any J £ [0, JQ\, any A, A contained in Z , any ui £ Z , any f such that 5 / fl A C A and |A| < n, we have EA>»(f In E£{p)) * The proof gives Cn = C exp Kn.
< Cn £ A ' " ( | V A / | 2 ) .
(9.2.1)
153
Some log-Sobolev inequality for effective single spin phase
Here EA'" is the A w i exp-$ > dX A .
expectation
with
respect
to
the
measure
When A = A = {i}, this a log-Sobolev inequality for a single spin effective phase cj)j{t) = 4>{t) + JJ2k~j I* ~~ wfc|2- Under the assumption (9.1.1), this inequality is a direct consequence of Bakry-Emery argument applied to 4>j(t) = <{>(t) + J Ylk~j I* — wfc|2- The conclusion of the lemma seems much stronger than this uniform inequality for the family of single spin effective phases (f>j. P r o o f of t h e l e m m a For A C A we define the probability measure E%u on MA by ££'"(/) : = £ A ' < " ( / ® 1 A \ A ) .
(9.2.2)
What we need is finally a uniform Log-Sobolev inequality with a constant depending only of the cardinal of A : |A|. Let us change the notations by introducing Ju = 2Jd , Jij = -J
if i ~ j , Jij = 0 else .
(9.2.3)
The phase appearing in the density of this measure with respect to has the form $ A ' " ( X A ) = $AJ(XA)
+ y%"'J(XA)
.
dXA
(9.2.4)
with1,
$A-'(XA) := 5 > f o ) + Yl J*xixk • J6A
j,fc€A
Changing by a multiplicative constant exp 2(sup |s|)|A|, it is enough to treat the case of a measure where is replaced by and consequently uniformly strictly convex on M. We use now the notation for . For this we have just to control the Hessian of ^^M'J(XA) with respect to the variables X A . Here we have (up to an irrelevant multiplicative constant)
ex P *i^(X A ) := /(exp _ $ A \ A , / ( X A \ A ) ) (exp _ $ A , A \ A , W ( X A \ A ) ) dXA\A , tSee (1.2.1).
[
'
154
Uniform log-Sobolev
inequalities
with $A\A,/(XA\A):=
£
4>(xk)+
fc€A\A
£
J^S,-,
(9.2.6)
Jejxezj.
(9.2.7)
i,.j'eA\A
and $A,A\A,U^A\AJ
.=
J^ £€A\A,jeAUA
c
Here Zj = Xj if j S A and u>j if j G Ac. We observe also that 3>A>-f is uniformly strictly convex (also independently of J > 0). We treat the second term in (9.2.4) as a perturbation for J small enough. What is relevant here is the Hessian of vErA'<"' (XA). When computing this Hessian we get, for i, j € A, (Hess^)ij=
^
JikJjt Cov A\AiZ(xk,xt)
.
(9.2.8)
M€A\A
For estimating this Hessian one has to use the uniform decay of the correlations for EA\A'Z, that is the property that there exists C such that, for A;, I £ A \ A, we have I CovA\A>z(xk
, xt)\ < C exp-—d(k,l)
,
(9.2.9)
and one is reduced to the following upper bounds : E i j e A l&ll&l T,kjeA\A \Jik\ \JjA exp-±d(k,l) ex d fc J ^ E M € A \ A P - i ( ^ ) ( E i € A \ ik\ I6I)(E J € A \Jje\ 161) < const. ||J||? i0O ||e|| 2 . Here we have defined ||Jj|i j 0 0 by ||J||i,oo : = s u p ^ | J j f c | =4Jd j
.
k
For J small enough, this can not perturbe the strict convexity of our phase and one can apply the Bakry-Emery argument (see Theorem 7.2.1) to the phase ^ r A ' w ' (XA) (with strictly convex) for getting the log-Sobolev inequality. We have completed the proof of the lemma. Remark 9.2.2 The conclusions of the lemma are also true if, for fixed J > 0, one assumes :
The role of the decay estimates for log-Sobolev
inequality
155
• the single spin phase ^ is a superconvex phase at oo. This means that, for any m > 0, there exists Cm such that "(x) > m , Vx s. t. |x| > C .
(9.2.10)
• We have the property of uniform exponential decay : (9.2.9). The first property is for example satisfied in the case of the quartic model (x) = Xx4 +
9.3
ux2.
The role of the decay estimates for log-Sobolev inequality
The proof of Log-Sobolev inequality rests on a second lemma which says : Lemma 9.3.1 Let us assume that (8.1.15) and the conclusions of Lemma 9.2.1 are satisfied. Then there exists J0, C and c such that, for any J G [0, J0], A C 2Zd, u> G 2ft , i e A, and any f sufficiently regular \EA'"(f2;xi)\ < C |A| exp(-cd(i,
SA))EA>»(f2)i
( ^ • " ( | V / | » ) + EA>» (/^log
^ ( B * ^ ) ) ) *
( 9 3 1}
,
where A = SA is the relative lattice support of f in A. Proof We treat the case* when i £ A. First we can write § , with i e A and i 0 A, EA'"{f2 • Xi) = EA-" ( / 2 ; E^f'ixij)
,
(9.3.2)
•(•The case when i 6 A is much easier § Using
/n'xn» f(x')9(x',x")™p-3>(x',x"W(x') (•
, . (\
= Jn.xn» /(-') (
V
• dn"{x")
„S(x',x")exp-*(x',x")dM"(x")\
;
e x p
_^,^WI)
)«P-H*>,v"WW)
•W ) •
156
Uniform log-Sobolev
inequalities
where Z( = xe if i G A and zt = u^ if I £ A. This is called "conditioning" by the probabilists. The equation is also called DLR-equation (for Dobrushin-Lanford-Ruelle). The covariance can be rewritten, using a duplication^ of the variables, as
= (EA'" x EA'")((f2
- f2)(EA\sf>z{xi)
EA\sf'*(Xi))\
-
(9.3.3)
where zt = xe if t e Sf and zt = u>e if I $ A. Using a telescopic sum to interpolate, that is writing EA\s''z(xi)
- EA\sf'*{Xi)
= [ h\t) dt, Jo
with zt = (1 - t)z + tz and h(t) := EA\sf'z'(xi), \EA\s''*(xi)
(9.3.4)
one has
-EA\sf-*(xi)\ < s^P9j€S(sf)
Here 6(Sf) = {k£A\Sf\
\EA\st'e(xj;xi)\
^2\zk\kesf
zk\
(9.3.5)
d(k, Sf) = 1}.
The uniform decay of correlations for the Gibbs measure implies
\EA'»{p;Xi)\ < \exp(-cd(i,SA))J2
(EA'U x £ A ' W ) ( | / 2 - P\ \xk -xk\).
(9-3-6)
fceA
^If fi is a probability measure on SI, we can always write the covariance of two functions u and v in the form :
(u, v) = I Jn*n
(u(x) — u(x)) • (v(x) — v(i))
dfj.(x) • dn(£)
The role of the decay estimates for log-Sobolev inequality
157
Therefore, it remains to prove that there is a constant C such that for any A; in A (EA-U x EA>») (\f2 - f2\ \Xk V
xk\)
/ V 2
/
i
< CEA'«(f )i • (EA>»(\Vf\2) + EA>» ^nog[wSm))Y
(9 3 7)
•
First we apply Cauchy-Schwarz's Inequality (EA>U
x EA>») ( | / 2 - f2\ \xk - xk\) < (iEA>» x EA<»)((f - / ) 2 ) ) ([EA>« x EAn((f
h
+ f? (a* - *fc)2)) * • (9.3.8)
One can use first the trivial estimate (EA>U x EA'W) ( ( / - / ) 2 ) < 4E A ' W (/ 2 ) .
(9.3.9)
In order to bound the other term, we first observe that (EA'" x EAn
( ( / + / ) 2 (xk - xk)2)
< 4(EA'" x EA'») (f
{xk - xk)2) . (9.3.10)
We then use the following entropy inequality \/t > 0,
fi(uv) < - log (/x(exp(to))) + -/x(ulogu).
(9.3.11)
where fx is any probability measure and the function / is a density (/ > 0 and /x(/) = 1). This is the immediate consequence of the inequality uv
+ ev -u
,
(9.3.12)
(with u > 0) called Young's inequality. Applying this inequality, we get for k € A, (EA'» x EA'») (f2 (xk - xk)2) < \EA'»{f2) log ({EA>» x EA>») (expt(xk 2 +l^(/ log(s475y)).
-
xk)2)) (9.3.13)
One first notes that, by symmetry of the measure, we have {EA'U x EA>")(xk -xk)=0.
(9.3.14)
Uniform log-Sobolev
158
inequalities
Furthermore, one knows that log-Sobolev inequality holds for the measure dfi = Z^i, exp
-$A
and for functions depending only on the variable Xk with a constant which is uniform with respect to u> and this is also the case for the duplicate measure when restricted to function depending on Xk and #&. This was just indeed the statement of Lemma 9.2.1. As a consequence of Herbst's argument (cf Section 7.3 and particularly Remark 7.3.3, point 3). More precisely we consider the measure d/x := dEA,w(XA) <8> dEA,u(XA) with functions depending only on two variables Xk and Xk- One knows that for t sufficiently small, there is c > 0 such that uniformly in u) {EA<" x EA'") (exp t(xk - xk)2) < c.
(9.3.15)
Here we have also used the property (9.3.14). Therefore there exists c > 0 and t > 0 such that (EA>»xEA'»)(f2(xk-xk)2)
Combining the previous inequalities, one derives (9.3.7).
9.4
Second part of the proof of the log-Sobolev inequality
This is a recursion argument on the cardinal n = |A| of the set A C 2Zd. One introduces 7n
= sup{ 7 L 5 (A) | |A| < n} ,
(9.4.1)
where 7 L S ( A ) denotes, for any A, the best Log-Sobolev constant valid (if it is true) for any measure EA'U with UJ £ 1R . What we have finally to prove is the property sup7„ < oo .
(9.4.2)
n>l
To prove this property, we have to prove the existence of C such that 72n
, for n > C .
(9.4.3)
Second part of the proof of the log-Sobolev inequality
159
To this end, starting of an arbitrary A CC 7Zd with |A| < 2n and for some / > 0, we choose A0 such that Ao C A and max{|Ao|, |A \ Ao|} < n} and we define, with {ij}^Li an enumeration of A \ AQ : Aj = A 0 U { i i , . . . , ij} , j = 1,2,..., m,
Si = V^ln, Ajtk = { l € A,-, d(e,ij+1) < f } , k e IN, Sj,k = VEAiHf2) UkelN and Ailfc ^ 0, Sj,k = / if Aj.fe = 0 . Here we have omitted the reference to ui. The functions Sj a n d Sj,k are considered as functions on A (independent of the variables in Aj or Ajtk). The three following lemmas will be useful. Lemma 9.4.1 There exists a constant C such that EA
<7n£A(|VAo/|2)
(P^E^P))
(9.4.4)
+c
E\\VAXAJ\2)
+
\J\2J2EA 3=0
where Kjif)
= sup{( CovAj(S2
, xk))2
| k € Aj,d(k,ij+1)
< 1} .
(9.4.5)
Proof We first decompose the l.h.s. of (9.4.4) as the sum of two terms
EA (/ 2 In 1JL->j
= E\S2 In g ) + EA (/ 2 In • Af° £ (/£),
(9.4.6)
The first term on the r.h.s. can be estimated as follows. We first observe that, because Ao C A,
EA(S2 In L) = EA
(EA°(S2
In 4 ) ) •
(9-4.7)
Jo \ Jo J This is what we call the Markov's property. Using the recursion argument, we get
EA(S2\nfy
< 7 n EA (£ A °(|V Ao /| 2 )) = lnEA (|V Ao /| 2 ) .
(9.4.8)
Uniform log-Sobolev
160
inequalities
For the second term, we use the decomposition
EA
(f2 ln *4J)=tt EA (ffln $ - fhiln fhi) = Z?Jo EA (/? In f] - E ^ (/ 2 ) ln /? + 1 ) 1
A
= Er=o ^ (^
Aj+1
(9.4.9)
(/|ln^ 7 )) •
We now observe that sfjnAj+1
= {t,-+i}.
We can consequently use (9.2.1) with |A| = 1 and get the existence of C\ > 0 such that : EA'+>(f*\n-£-)
< Cx EA^
(|V^.+1//) .
(9.4.10)
Jj+i
We now use the following easy sublemma : Sublemma 9.4.2 There exists C such that, for any A CC 2Z , for all £ $. A, we have
\VXe^EHP)\ <£ A (|V X £ /| 2 ) 5 +CU|su P { i€A , d(i,e)
(9.4.11) This gives for a suitable uniform constant C EA^(ffln^-) Jj+i
< C [EA^ \
( | V X . + 1 / | 2 ) + \J\2^fl-) h
.
(9.4.12)
)
From (9.4.9) and (9.4.12), we finally obtain
EA (/
2in
^4j) - c (EA ( | V A \ A ° / | 2 ) + ' j | 2 %EK (^jf
(9.4.13) By (9.4.6), (9.4.7) and (9.4.13), we get (9.4.4) and consequently the proof of Lemma 9.4.1.
Second part of the proof of the log-Sobolev inequality
161
Proof of the sublemma We have
= \\E\pyh
(2E\fvxj)
+ j E {i€ A,^ } # A (/ 2 ; *<)) I-
We now use Cauchy-Schwarz for the first term of the r.h.s. and get
\vxeJW(p)\ < ±EHf2)-?(2E\f)iEH\VxJ\2)l+2\J\dSnp{ieA^e} = E\\VXJ\*)2
+ \J\dEA(p)~i
\EHf2^i)i)\
s u P { i e A i i ^ } \E\f;
Xi)\.
The sublemma is proved. Lemma 9.4.3 There exists a constant C such that ^\7v/)
•fl h
2
ti
2 KlA, / ITT , r2 l_ < ^l\~^ c £ e X p ( I- £"-) \ E* | V A , t + I /C\1 | 2 +, /J-2l l n/ ^ + / l n - ^/ J T fc=0 ° V 3,k 3, k+i J (9.4.14)
^
Proof Suppose that £ € Aj and d(£, ij+i) < 1. We then have that
(E^(fxe))2 = (£ A '((/ 2 -/?)s<)) 2 = (EL0°O(^((/^-/^+1)^)))2
< (ElS(* +1)- 2 )
(E£S(*
+1)2 (E^E^^ifl;
^))2). (9.4.15)
where we have used Cauchy-Schwarz for getting the last line. Using (9.3.1), we obtain, observing also that k d(£, A j i k + 1 \ A iifc ) > - - 1 ,
\EA>-k+1 (f*k ; xt)\ < Cexp- (J£) / ^ ( J * + / | ) ,
(9.4.16)
162
Uniform log-Sobolev
inequalities
where
h:=EA^(\VAj,k+1fjik\2), and I 2 :=jE^.*+i
/j>. in
>i,k
/ j,k+l,
Using (9.3.1) and Sublemma 9.4.2, we get easily (with the same notations as in Lemma 9.3.1) the following sublemma : S u b l e m m a 9.4.4
+ C |A| e x p ( - ^ ) (V(|V A /P))* + (EA (/"in ( 5 * ^ ) ) ) ' (9.4.17) We now apply the sublemma with A replaced by Aj^ and I € A^fc+i \ A^fc. Taking then the square of the inequality and the expectation on Ajfc+i>w e get, using also the property EA^k+1EA^kg = EA^k+1g, h
+ l)Ad
EA-k^
(|V A ., fc+I /| 2 ) + EA-k^
/2ln
11 j,k,
(9.4.18) For the control of Ii we use simply the Jensen Inequality'!, which gives I2<EA*-k^
/2ln \
P .. Jj,k+1 J
(9.4.19)
From (9.4.16), we get
EA*(EA^(flk;xe))}2 < C exp-A [EAI
((fjM1(l}+lfy)
(9.4.20)
<2Cexp-A/2(/1+/2); using Cauchy-Schwarz in the last line.
II One can also more directly use the positivity of the entropy recalled in (7.1.4).
Second part of the proof of the log-Sobolev inequality
163
We then obtain, putting (9.4.18) and (9.4.19) and (9.4.20) together,
< C exp - £ f] • E^
(|V Aj , fc+1 /| 2 + / 2 In £- + f I n - £ - ) . (9.4.21)
Plugging this into (9.4.15), we get the following bound (gAi(/a;xt))3
/?
< C E ^ e x p - ^ J5Ai (|V Aj , fc+1 /| 2 + / 2 l n £ - +fHnjA-)
. (9.4.22)
This ends the proof of Lemma 9.4.3. Lemma 9.4.5 There exists a constant C such that, for fco = 1, 2, • • •, [n^] — 1, there exists a constant D, such that E ^ f l n ^ ) < lnEA (|V A o /| 2 )
+7n
C e x p ( - j £ ) EA (|V A /| 2 )
2
2
+DE* (|V A /| ) + C e x p ( - ^ ) E* ( / In ^
(9.4.23) )
.
Proof We have seen from (9.4.14) that
""W^/)' E^ i-o v n < fc=0
\
^ /
J = 0
\
•'j.fc
Jj,k+l ',fe+l/
(9.4.24) Let us first note that, if (A; + l ) d < n,
EA ( / \
2
ln^ + / J
j,k
2
l n - ^ ) Jj,k+1 J
< 2 7 ( f e + 1 ) d E A (|V Ai , fc+1 /| 2 ) .
(9.4.25)
164
Uniform log-Sobolev
inequalities
This comes from the Markov Property and the recursion argument. We observe indeed that (k + l)d < n and that : EA ( / 2 l n £ ) = EAEA^
(/2ln^)
(9.4.26) .
On the other hand we have, without any condition on k,
£A /2l
+/2l
( "te) "(^))
£2£A /2,
( "(s4)))' (9.4.27)
For the bound (9.4.27), we observe that EA ( / 2 In £ )
= EA (/ 2 In / 2 ) - EA ( / 2 l n ( / 2 , ) ) = EA(p\np)-EA{flk\n(flk)) < EA (/•* In f•*) - EA (f2) In EA(f2) ,
where we have used Markov's property in the second line, and, in the last line, Jensen's Inequality (or (7.1.5) with g = EA^k(f2)) and Markov's property. Let us come back to the analysis of the r.h.s. of (9.4.24) that we decompose as the sum of three terms T(0,fco— 1) + T(ko, k\ — 1) + T(ki, +oo) corresponding to the decomposition : Y,t=o = Efcl"^1 + EfcL"^ + E t T t , . with &i = [rid] — l. We first estimate T(0, k0 - 1) using (9.4.25) ko — l m — 1
T(0,ko-l)
Y,
E
e x
j
P - c
( 1 + 27
(fc+1)d)^A(|VA^/|2) •
fc=0 j=0
(9.4.28) We now resum the r.h.s. in the form T(0,*«,-1)
+
1 2 7 ( f c + 1 ) ,) ( Y \{j s.t. eeAj.k} (9.4.29)
Second part of the proof of the log-Sobolev
165
inequality
We need now to estimate the cardinal of {j s.t. £ G A^fc} from above by a constant depending only on k. But for a given £, the cardinal is immediately seen as bounded from above by (fc + l ) d . We consequently get, with a new constant C T ( 0 , f c 0 - l ) < C J > A ( | V ^ / | 2 ) ] T e x p - ^ ( l + 27(fc+1)<0. (9-4.30) ^6A
fc=0
We have finally obtained, for ko < [n*)] — 1, T(0, k0 - 1) < C(fco) ^ A ( | V A / | 2 ) ,
(9.4.31)
with /feo-i
C(fco) = C
,
\
E e x p - - ( 1 + 2 7 ( f c + 1 ) .) \fc=0
.
(9.4.32)
/
By a similar argument, we get for the second term T(k0,h
- 1) < C(l + 2 7 n ) e x p - ^ S A ( | V A / | 2 .
(9.4.33)
Finally for the first term, we simply use (9.4.27) and get T(fci,+oo)
(9.4.34) and finally, taking account of the choice of fci and of the definition of m < n, we obtain, for a possibly new larger C, T(fc1)+oo)
A
(/2ln(^5y))
.
(9.4.35) We then get the conclusion of the lemma from (9.4.4), (9.4.24), (9.4.31), (9.4.33) and (9.4.35). End of the proof of the inequality (9.4.3) We recall that we need to prove the inequality for n large enough. We shall also use the finiteness of 7„ for n < no which is a consequence of
166
Uniform log-Sobolev
inequalities
Lemma 9.2.1. We apply Lemma 9.4.5 with Ao and with A \ Ao. Taking the half sum, we get E
A
( /
2
l n ( ^ ) )
<7n((I+Cexp-f)
£A(|VA/|2)
+CC(k0) £ A ( | V A / | 2 ) + C e x p - ^ EA ( / 2 l n ( ^ l ) ) ) • (9.4.36) This equation is valid for any pair (feo,n) such that 0 < fco < [n 3 ] — 1. We first choose fco such that 1
. n
2
k
° <-
3
Z
0
We get that (9.4.36) is satisfied if n > (fco + l ) d • We can now compute C(fco) through (9.4.32) and, for n such that 1-Cexp-%->!, n > (fc0 + l)d ,
(9.4.37)
we obtain EA
( f2H-E^))
< \ln £ A ( | V A / | 2 ) + ^CC(fco) S A ( | V A / | 2 ) . (9.4.38)
Remark 9.4.6 What we have actually proved is some variant of Zegarlinski's theorem in the following form : If we have the properties (8.1.15) and (9.2.1), then we have the uniform log-Sobolev inequalities.
9.5
Conclusion
These results can be completed by the following theorem which establishes the equivalence in the case of a superconvex phase of the various properties. Theorem 9.5.1 Under the assumption that the single-spin phase is superconvex at oo, that is that, for any m > 0, there exists a bounded C2 function s such that, (4> +
s)"(t)>m,
(9.5.1)
Notes
167
then the following conditions are equivalent : (1) Correlations decay exponentially fast uniformly with respect to A, LO in the sense of (8.1.15). (2) The Poincare inequalities hold uniformly with respect to A,o>. (3) The log-Sobolev inequalities hold uniformly with respect to A, w. We do not give here the proof of this theorem, but note that this last theorem does not say if the log-Sobolev inequality holds uniformly or not. This is actually true when d = 1 and for J small enough for a general d. It just establishes that this property is equivalent to simpler properties.
9.6
Notes
Section 9.1 • The main object of this chapter is to give a complete proof of Zegarlinski's statement [Ze] which was left as an exercise by the author who treats in detail the case d = 1. We do not succeed completely because we can only treat the perturbative case and are for example unable to recover the case d = 1 by our approach (See however Remark 9.2.2). • We actually follow and improve Yoshida's proof [Yoshl] along the lines of our work with T. Bodineau [BodHel] by using our refined decay estimates [He2l]. Yoshida was indeed treating only a particular case. • The theorem stated by B. Zegarlinski [Ze] is a little different in the conclusions. We shall prove it under a stronger assumption by assuming no restriction on the size of Ao. • More recently M. Ledoux [Led2] presents also another elegant proof in Bakry-Emery spirit and let us also mention the course by G. Royer [RoyG2], who in particular mentions his former paper [RoyGl]. • Other extensions based on a result of [BobGo] are given in [Ge]. Section 9.2 • Here we follow Yoshida's approach [Yoshl].
168
Uniform log-Sobolev
inequalities
Section 9.3 • These decay estimates were already present in the proof of the previous lemma (but not in a decisive way). They were introduced in the context of the log-Sobolev inequalities by B. Zegarlinski. N. Yoshida replaced them by a stronger mixing condition. We show here that this is actually not necessary. • For technical reasons, N. Yoshida was also limited to the case of the model** and does not obtain Zegarlinski's theorem in full generality. Section 9.4 • Here we refer for the end of the proof to the presentation of Yoshida [Yoshl] that we have followed rather closely using also suggestions of T. Bodineau. • Roughly speaking, N. Yoshida's proof of this second part is essentially the transposition to the continuous case of a proof established in some discrete case by Lu and Yau [LuYa]. Section 9.5 • Theorem 9.5.1 is a part of a more elaborate theorem established in [Yosh3] but with additional conditions. The proof of the statement given here can be found in [BodHe2].
**In [Yoshl], some extension is obtained (conditions U l and U2) but some superquadratic increase is in particular assumed. For Polynomials <j> : 4>{t) — ^ " L , a„t2", N. Yoshida imposes for example the conditions m > 1, am > 0 and a„ > 0 for v > 1.
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Index
De Rham Complex, 9, 10 decay estimates, 147, 155 decay of correlations, 5, 34, 133, 142 differential quotients method, 98 Dirichlet form, 16, 105 Dirichlet problem, 82, 92 Dirichlet realization, 87 Discrete spectrum, 83 distorted exterior differential, 13 DLR-equation, 156 double well case, 33 duplication, 156
adjoint operator, 78 Agmon estimates, 68 approximate eigenvectors, 58 approximate solution, 59 Bakry-Emery argument, 118, 153 Betti number, 15 Bochner Laplacian, 21 Bochner-Lichnerowicz-Weitzenbock formula, 17 Boltzmann constant, 23 bottom of the spectrum, 93, 141 Brascamp-Lieb inequality, 21, 22, 147
eikonal equation, 60, 62, 75 entropy, 101 entropy inequality, 157 essential spectrum, 83, 84 essentially selfadjoint operator, 86 exterior differential, 9
closable operator, 77 closed extension, 78 closed graph Theorem, 77 closed operator, 77 compact operator, 36, 94 compact resolvent, 82 conditioning, 156 convex case, 33 convexity, 151 convexity at infinity, 118 correlation function, 27 correlation length, 27 covariance, 2, 4, 17, 20, 44, 151 critical temperature, 29 Curie-Weiss model, 23, 33 cylindrical functions, 43
ferromagnetic, 28 first eigenvalue, 19, 63 first eigenvector, 19 formally symmetric operator, 36 Fredholm theory, 19 free boundary condition, 134 free boundary model, 3 free energy, 26 free energy per spin, 28 free Laplacian, 88 177
178 Friedrichs extension, 16, 91, 98 functions of a selfadjoint operator, 81 Gibbs measure, 147, 149 globally hypoelliptic operator, 108, 131 ground state, 19 groundstate energy, 19 Hardy inequality, 132 harmonic approximation, 15, 55, 63, 67, 74 harmonic Kac operator, 51, 73 harmonic oscillator, 51, 63, 89 Herbst's argument, 115, 158 Hering formula, 70, 75 Hermite functions, 54, 63, 83 Hermite polynomial, 64 Hilbert-Schmidt condition, 39 Hilbert-Schmidt operator, 37 Hodge theory, 15 hyperbolic map, 62, 73 hyperbolic point, 60 IMS formula, 74 interaction energy, 28 interaction phase, 138 interaction potential, 34 Ising model, 25, 28 Kac conjecture, 50 Kac operator, 39 Kato Theorem, 89, 98 Krein-Rutman Theorem, 26, 40, 41, 49, 106 Lagrangian manifold, 60 Laplace Beltrami, 83 Laplace integral, 1, 4, 24 Laplace method, 6, 31, 56, 59, 114 Laplace-Beltrami operator, 12, 98 largest eigenvalue, 7, 29, 74 lattice approximation, 49 lattice support, 135, 155
Index Lax-Milgram Lemma, 16, 91 log-Sobolev inequality, 2, 101, 102, 117, 133 lowest eigenvalue, 15, 19 magnetic field, 23 Markov's property, 159 max-min principle, 93 maximal extension, 78 Maximum principle, 55 mean field theory, 23 mean value, 1, 43 Mehler formula, 73 minimal extension, 78 minimax Principle, 72 Morse function, 14 Morse inequalities, 21 Morse Lemma, 32 nearest-neighbor, 3, 28 Neumann realization, 87 one-particle phase, 3, 133 one-well problem, 63 pair correlation, 4, 44 partition function, 24, 28, 29 Pauli matrix, 27 periodic case, 34 periodic model, 3 periodic sublattice, 45 Perron-Frobenius Theorem, 26, 41, 42,53 phase transition, 5, 34, 35, 49 Planck constant, 15 Poincare inequality, 2, 20, 104, 117, 118 positivity improving property, 106 positivity preserving property, 106 precompactness, 94 quasimode, 71 resolution of the identity, 79
179
Index Riesz Theorem, 78, 88 Schrodinger operator, 15, 19, 51, 62, 77 Schrodinger operator with constant magnetic field, 84 Schur's Lemma, 49, 66, 138 selfadjoint operator, 78 semi-classical analysis, 15, 51 semi-classical limit, 34 semi-classical parameter, 34 semi-group, 105 single spin effective phase, 153 single-spin phase, 142 Sobolev inequality, 107 spacing, 29 spectral decomposition Theorem, 78, 80, 97 spectral Theorem, 58 spectral theory, 77 spectrum, 58, 80 speed of convergence, 35, 43 spins lattice, 23 splitting, 27, 29, 35, 43, 70, 74, 75, 147 stable manifold Theorem, 62, 73 stationary phase theorem, 6, 31 statistical mechanics, 6, 133 statistical mechanics with discrete spins, 23 strictly convex case, 75, 103 strictly positive kernel, 40 superconvex phase, 155, 166 symmetric double well, 65, 73
symmetric symmetric symplectic symplectic
kernel, 36 operator, 36, 77 geometry, 73 map, 60
temperature, 23, 31, 34 thermodynamic limit, 7, 24, 28, 42, 46, 50 thermodynamic properties, 133 trace, 6 trace class condition, 39 transfer matrix, 26-28, 46 transfer matrix method, 34, 49 transfer operator, 4, 7, 51 Trotter-Kato formula, 75, 106 tunneling, 29 unbounded operator, 77 uniform decay estimate, 128, 135 uniform log-Sobolev inequality, 151 uniform Poincare inequality, 6, 142 uniformly strictly convex, 20 variance, 2, 20, 45 Weyl's Theorem, 85 Witten Complex, 13 Witten Laplacian, 4, 9, 21, 92, 95, 108, 136, 142 WKB construction, 59 WKB solution, 62 Young's inequality, 157
Series on Partial Differential Equations and Applications-Vol. 1
Semiclassical Analysis, W i t t e n Laplacians, and Statistical
Mechanics
his important book explains how the technique of Witten Laplacians may be useful in statistical mechanics. It considers the problem of analyzing the decay of correlations, after presenting its origin in statistical mechanics. In addition, it compares the Witten Laplacian approach with other techniques, such as the transfer matrix approach and its semiclassical analysis. The author concludes by providing a complete proof of the uniform log-Sobolev inequality.
ra, h)~l exp-
ISBN 981-238-098-1
www. worldscientific. com 5049 he
