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Transfer operator, WKB-analysis
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\topmatter
\title{The low-temperature limit of transfer operators in fixed
dimension}\endtitle
\author{Jacob Schach M{\o}ller\footnote{Supported by TMR
grant FMRX-960001\newline}}\endauthor
\affil Universit{\'e} Paris-Sud\\
D{\'e}partement de Math{\'e}matique\\
Orsay, France\\
e-mail: Moeller.Jacob\@math.u-psud.fr
\endaffil
\abstract
We construct the $0$'th order low-temperature WKB-phase for the first
eigenfunction of a transfer operator in a large
domain around a non-degenerate critical point for the potential.
The $0$'th order low-temperature phase is shown to solve the eikonal
equation in the strong-coupling limit and we obtain non-local estimates
on the $0$'th order phase, which are preserved in the limit.
We furthermore use the IMS localization technique to study the
two highest eigenvalues of the transfer operator
in the case where $V$ is allowed to have many non-degenerate global
minima.
\endabstract
\endtopmatter
\document
\head{Part I: Introduction}\endhead
\subhead{Section I.1: Short presentation of the results}\endsubhead
This paper is concerned with three problems related to
the low-temperature limit of transfer operators. A transfer operator
is a bounded positive operator on $L^2(\Bbb R^m)$ with integral kernel
$$
K(x,y) = (\beta J)^{\frac{m}2}e^{-\frac{\beta}2 V(x)}
e^{-\frac{\beta J}2 |x-y|^2}e^{-\frac{\beta}2 V(y)},
$$
where $\beta$ is the inverse temperature, $J$ is a coupling constant
and $V\in C^\infty(\Bbb R^m)$ is a non-negative potential. We will always
assume $J>0$ (ferromagnetism), $V$ has a finite number of
non-degenerate global minima (where $V$ equals $0$) and $\inf_{|x|>R}V(x)>0$,
for some $R>0$. One can also consider other dispersion relations than
$|x-y|^2$, for example $\omega(x-y)$ where $\omega$ is some strictly convex
function.
The transfer operator can be viewed as a continuous spin analogue of
the transfer matrix, which plays a central role in the study of the
Ising model. It was used by Helfand and Kac in [HK] to treat
the mean-field limit of some discrete spin-systems. The mean-field
parameter becomes a ferromagnetic coupling constant for the corresponding
transfer operator. See [K] for an exposition. The study undertaken here,
by analogy with the Ising model, relates to the statistical mechanics of
continuous spin-systems. See [F] and [He5] for details. The relation
is sketched for spin-chains in Section III.5.
There are two limits which are of a semiclassical nature. The strong-coupling
limit (SC) $J\to + \infty$ and the low-temperature limit (LT)
$\beta\to +\infty$. For $\beta$ or $J$ sufficiently large
at least one eigenvalue will appear at the top of the spectrum of $K$,
which can be viewed as the inverse of the spectrum of a Schr{\"o}dinger
operator since
$$
K = (2\pi)^{\frac{m}2}e^{-\frac{\beta}2V}e^{\frac1{2\beta J}\Delta}e^{-\frac{\beta}2V}.
$$
The eigenvalues will be numbered in decreasing order making the first
eigenvalue the highest. We furthermore note that, by the Perron-Frobenious
Theorem, the first eigenvalue is non-degenerate and its eigenfunction can be
chosen strictly positive.
We will in this paper be interested in the following
problems
{\it 1) A WKB-construction, in large domains, of the first eigenfunction
of $K$ in the low-temperature limit.}
In [He3] and [He4] Helffer showed how to make WKB-constructions
for the first eigenfunction of $K$ in the low-temperature and strong-coupling
limits. A semiclassical expansion was obtained for the logarithm of
the first eigenfunction
and we write $\varphi_0^{(LT)}$ and $\varphi_0^{(SC)}$ for the two $0$'th
order phases (see Sections I.3 and I.4). The construction of $\varphi_0^{(LT)}$ relied on an application
of the local stable manifold Theorem and in this paper we give a more
detailed analysis of the problem. We construct $\varphi_0^{(LT)}$
in certain 'large' domains and provide estimates which are of a non-local
nature.
{\it 2) Study the strong-coupling limit of $\varphi_0^{(LT)}$ and its relation
with $\varphi_0^{(SC)}$.}
The $0$'th order strong-coupling phase $\varphi_0^{(SC)}$ satisfies the eikonal equation
$$
|\nabla \varphi|^2 = V,
$$
and hence can be described as the generator of the outgoing stable manifold
of the Hamiltonian flow generated by $H(x,\xi) = \frac12\xi^2 - V(x)$.
On the other hand $\varphi_0^{(LT)}$ is the generator of the stable incoming
manifold for a discrete dynamical system on phase-space (parametrized by $J$)
and we show that in the strong-coupling limit this dynamical system approximates
(when rescaled) the continuous dynamical system given by the Hamiltonian flow of
$H$. This connection enables us to get non-local estimates on solutions
to the eikonal equation in 'large' domains as a byproduct of corresponding
estimates on the $0$'th order low-temperature phase. We note that the eikonal
equation enters at leading order in the WKB-analysis of the Schr{\"o}dinger
operator and in semiclassical expansions of Laplace integrals. See [He1],
[Sj1] and [Sj2].
The analysis in 1) and 2) can be done in the framework of standard functions
(control with respect to dimension), see [Sj2], which will be the topic
of future work.
{\it 3) Use localizations to study the full transfer operator when $V$
has many non-degenerate global minima.}
We use an IMS type localiztion technique, which is similar (but simpler)
to the one employed in [Si] to analyze the first two
eigenvalues of the Schr{\"o}dinger operator in the semiclassical limit.
We determine the logarithm of the first eigenvalue, using the expansion coming
from the WKB-analysis, up to an $O(h^\infty)$ error. As for the splitting
between the two first eigenvalues we consider two cases. For a symmetric
double well potential we show that the splitting vanishes exponentially in
the low-temperature limit. In the case of a unique global minimum we show that,
up to an exponentially small error, the splitting is given by the splitting
of the localized problem, which can be treated by harmonic
approximation (see [He5]).
\subhead{Section I.2: Overview and acknowledgements}\endsubhead
The paper is divided into three parts. The present part is
an introduction, which is primarily concerned with an exposition
of the WKB-construction for the highest eigenvalue of the transfer
operator in the low-temperature and strong-coupling limits,
and the corresponding eigenfunctions. In Section I.3 the crucial
step of determining the low-temperature WKB-phase to $0$'th order is explained
and in Section I.4 the corresponding step for the strong-coupling limit
is recalled. A relation between these two limits is exploited in
Section I.5 to indicate how one can obtain solutions to the eikonal
equation as limits of $0$'th order low-temperature phases.
In particular we get global solutions
if the potential is globally convex.
Part II of the paper is concerned with the construction of
the WKB-phases based on the characterizations derived in Part I.
In Section II.1 we construct the $0$'th order low-temperature
phase and provide bounds on its Hessian.
In Section II.2 we construct a Hamiltonian
flow which we use in Section II.3 to obtain solutions to the eikonal equation
following the idea outlined in Section I.5. We give some extra
results on the $0$'th order low-temperature phase and the solution to the
eikonal equation in Section II.4.
The last part of the paper connects the expansion obtained in Part II
with the problem of determining the first eigenvalue of the
transfer operator. In Section III.1
we show that the restriction of the transfer operators to neighbourhoods
of local minima of the potential contain all the low-temperature information.
In Section III.2 we prove that the first eigenvalue of the restricted
tranfer operator is correctly described by the WKB-construction and we analyze
the second eigenvalue in the case of a unique global minimum.
In Section III.3 we comment on global constructions in the case
where the potential is globally convex. In III.4 we discuss the consequences
of the results for classical spin-chains and we compare with the recent results
of [BJS].
The references to work related to WKB-constructions for Schr{\"o}dinger
operators are chosen based on relevance to control with respect to
dimension and are not intended to represent the subject as a whole.
Studying the low-temperature limit of the transfer operator
using semiclassical techniques was suggested to the author by B.\,Helffer.
We would like to thank V.\,Bach and T.\,Ramond for
discussions and B.Helffer for valuable comments and for his help in rooting
out errors and misprints.
\subhead{Section I.3: The low-temperature limit}\endsubhead
Throughout the paper we will for the sake of brevity use the notation
$f^\prime$ for $\nabla f$, the gradient of $f$. In the case
where $f$ depends on more than one variable we will
write $(\nabla_x f)(x,y)$ in order to avoid confusion.
The same notation will be used for Hessians.
In the introduction we simplify the exposition by assuming that the
potential $V\in C^\infty(\Bbb R^m)$ is globally convex
$V^\dprime>0$ and has a unique global minimum at $0$
$$
V(0)= V^\prime(0)=0.
$$
We define the transfer operator $K$ by its kernel
$$
K(x,y) = (\pi h)^{-\frac{m}2}e^{-\frac1{2h} V(x)}e^{-\frac{J}{2h}|x-y|^2}
e^{-\frac1{2h} V(y)}.
$$
Notice that we have replaced the inverse temperature by a
semiclassical parameter $h=\beta^{-1}$ and chosen a more natural prefactor
for the limit considered.
In order to iteratively construct semiclassical expansions for the
first eigenvalue of the transfer operator in the low-temperature
limit we make the following ansatz for the first eigenfunction
$$
\psi_1(x;h)= e^{-\frac1{h}\varphi(x;h)},\qquad \varphi(x;h)\sim
\sum_{k=0}^\infty \varphi_k(x)h^k,\tag{3.1}
$$
where $\sim$ will denote formal expansions.
This type of ansatz was used for Schr{\"o}dinger operators in
[Sj1] instead of the more traditional ansatz,
$\psi_1 = ae^{-\frac1{h}\varphi_0}$,
where the amplitude $a$ is a formal power series in $h$.
The reason is that the control of terms with respect to dimension
is difficult with the latter approach.
As for the eigenvalue we suppose
$$
\lambda_1(h)=e^{-F(h)},\qquad F(h)\sim \sum_{k=0}^\infty F_k h^k.\tag{3.2}
$$
The idea is to determine $\varphi$ and $F$ inductively by requiring
that the integral below is equal to $1$ or rather $e^{O(h^\infty)}$.
$$
e^{F(h)}\int_{\Bbb R^m}
e^{\frac1{h}\varphi(x;h)}K(x,y)e^{-\frac1{h}\varphi(y;h)}dy.\tag{3.3}
$$
The strategy is to make a coordinate transformation to bring the integrand
into a certain form depending on the scaling of $h$.
We consider a change of coordinates $y\rightarrow \xi(y;x,h)$,
where $\xi$ has a formal expansion, which gives the integral (3.3)
above the form
$$
(\pi h)^{-\frac{m}2}\int_{\Bbb R^m} e^{-\frac1{h}\xi^2(y;x,h)
+\ln\det\nabla_y\xi(y;x,h)}dy.\tag{3.4}
$$
This will imply that $F(h)$ is formally an expansion of the logarithm of
the first eigenvalue. We make the further assumption that $\xi$ is the
gradient with respect to $y$ of a function, in order to make its
derivative symmetric. That is we look for
$$
\xi(y;x,h) = \nabla_y f(x,y;h),\qquad f(x,y;h) \sim
\sum_{k=0}^\infty f_k(x,y)h^k.\tag{3.5}
$$
To ensure that the determinant of the Hessian of $f$ is positive
(at any finite order) for small $h$ we make the assumption that $f_0$,
the coordinate change to first order, is a convex function of $y$
for any $x$.
Using the explicit form of the integral kernel of the transfer operator
we find the equation between formal power series
$$
\align
\varphi(y;h)-\varphi(x&;h) +\frac12(V(x)+V(y))+\frac{J}2|x-y|^2 \\
&= |\nabla_y f(x,y;h)|^2
- h\ln\det\nabla_y^2f(x,y;h) + h F(h).\tag{3.6}
\endalign
$$
Consider the $0$'th order equation
$$
\varphi_0(y)-\varphi_0(x) +\frac12(V(x)+V(y)) +\frac{J}2|x-y|^2 =
|\nabla_y f_0(x,y)|^2.\tag{3.7}
$$
Notice that we have to determine the two unknowns $\varphi_0$ and $f_0$ at
the same time. Fix $\varphi_0(0) = 0$ (can be chosen freely).
By the assumption that $f_0$ is convex in $y$ for any $x$ we find that
for any $x$ there exists a unique critical point $\Phi_J(x)$ at which
$\nabla_y f_0$ vanishes. Note that the gradient of the right hand side
with respect to both $x$ and $y$ vanishes at the critical manifold
$(x,\Phi_J(x))$. This gives the following equations (for $\Phi_J(x)$)
$$
\align
\varphi_0^\prime(y) +\frac12V^\prime(y) + J(y-x) &= 0\\
-\varphi^\prime(x) +\frac12 V^\prime(x) + J(x-y) &=0.\tag{3.8}
\endalign
$$
Notice that given $(x,\varphi_0^\prime(x))$ we can determine
$(\Phi_J(x),\varphi_0(\Phi_J(x)))$.
Inspired by this observation we introduce a diffeomorphism
of the symplectic space $\Bbb R^m_x\times\Bbb R^m_\xi$
$$
\kappa(x,\xi;J) = (\kappa_x(x,\xi;J),\kappa_\xi(x,\xi;J))=(y,\eta),
$$
given by the set of equations
$$
\align
\eta + \frac12 V^\prime(y) + J(y-x) &= 0\\
-\xi + \frac12 V^\prime(x) + J(x-y) &= 0.\tag{3.9}
\endalign
$$
The $0$'th order phase $\varphi_0$ will now by (3.8) satisfy
the relation
$$
\kappa(x,\varphi_0^\prime(x);J) =
(\Phi_J(x),\varphi_0^\prime(\Phi_J(x))).\tag{3.10}
$$
In other words the graph of $\varphi_0^\prime$ is a stable manifold
for the fixed point $(0,0)$ of the discrete dynamical system $\kappa$.
One can see by convexity that the critical point $\Phi_J(x)$
will be closer to $0$ than $x$
(if $\varphi_0$ is convex) which indicates that the graph is the
incoming manifold.
One can check that $(0,0)$ is a hyperbolic fixed point and that $\kappa$
is a symplectic transformation which implies that the stable manifolds
are Lagrangian submanifolds of $\Bbb R^{2m}$.
From (3.7) one can also see that the $0$'th order phase will be
a fixed point of the transformation
$$
\tilde{T}\varphi(x) = \frac12 V(x)
+ \inf_{y\in\Bbb R^m}(\frac12 V(y) +\varphi(y)
+\frac{J}2|x-y|^2),\tag{3.11}
$$
which turns out to be a contraction on convex functions.
This observation will be the key to the global construction given in
Section II.1. Notice that the stable manifold Theorem always
gives a local construction. This approach was taken
in [He3] and in [He4] control with respect to dimension is achieved.
Iterating the transformation in (3.11) starting with the
function $\varphi=\frac12 V$ gives a pointwisely non-decreasing
sequence which is bounded from above. Hence it converges to some
function which is locally Lipschitz. This is true for any positive potential
but we lack explicit control of the limit. See [He2], [He3] or [He5].
\subhead{Section I.4: The strong-coupling limit}\endsubhead
This limit has been treated by Helffer in [He4] and by Helffer and Ramond
in [HeRa] and we present here
the results. The semiclassical parameter will be
$$
h = J^{-\frac12}.
$$
We put $\beta=1$ (or alternatively transform it into the potential).
One can proceed here as in the previous section and look
for a suitable change of coordinates. The scaling is however different
and one should aim at bringing the integral (3.3) on the form
$$
(\pi h^2)^{-\frac{m}2}
\int_{\Bbb R^m} e^{-\frac1{h^2}|\xi(y;x,h)|^2+\ln\det\nabla_y\xi(y;x,h)}dy
$$
instead of the form (3.4). (Notice that the transfer operator
in this limit has an extra factor of $h^{-\frac{m}2}$.)
This will give the equation (between formal power series)
$$
\align
h\varphi(y;h)- &h\varphi(x;h)+\frac{h^2}2(V(x)+V(y))+\frac{J}2|x-y|^2 \\
&= |\nabla_y f(x,y;h)|^2
- h^2\ln\det\nabla_y^2f(x,y;h) + h^2 F(h).\tag{4.1}
\endalign
$$
Analyzing this one finds that the $0$'th order WKB-Phase
$\varphi_0$ must satisfy the eikonal equation
$$
|\varphi_0^\prime(x)|^2 = 2V(x).\tag{4.2}
$$
One can also describe $\varphi_0$ in way more parallel to what was
done in the previous section. Notice that $(x,\varphi_0^\prime(x))$ lies
on the zero-energy manifold for the Hamiltonian
$$
H(x,\xi) = \frac12\xi^2 - V(x).\tag{4.3}
$$
In fact $(x,\varphi_0^\prime(x))$ will be the outgoing stable manifold
at the hyperbolic fixed point $(0,0)$
for the continuous symplectic dynamical system given by
the Hamiltonian flow $\Psi_t(x,\xi)$ for the Hamiltonian above
$$
\align
\dot{\Psi}_t(x,\xi) &= (\nabla_\xi H)(\Psi_t(x,\xi))\\
\dot{\Psi}_t(x,\xi) &= -(\nabla_x H)(\Psi_t(x,\xi))\\
\Psi_0(x,\xi) &= (x,\xi).\tag{4.4}
\endalign
$$
\subhead{Section I.5: A connection between the two limits}\endsubhead
In this section we will explain a central point of this paper
which links the low-temperature limit with the strong-coupling limit.
Suppose we take the low-temperature limit and then subsequently ask
what happens when the coupling becomes large. We can on a formal level think
of this as replacing $J$ by the semiclassical parameter $h^{-2}$
which was chosen to be $\beta^2$. The scaling is now $h^{-1}$ on the potential
and $h^{-3}$ on the coupling term. If one tries to construct a $0$'th
order phase with this scaling one finds exactly as in the previous section that
it must satisfy the eikonal equation and that it enters as a $h^{-2}$
term. Now the $0$'th order low-temperature phase depends on $J$ and
entered on the $h^{-1}$ level. In order to
get the scaling $h^{-2}$ it is natural to suggest that
$J^{-\frac12}\varphi_0^{(LT)}$
should scale as a constant for large coupling constants $J$.
In fact it not only scales as a constant
it converges to the strong-coupling $0$'th order phase. In other words
$$
\lim_{J\rightarrow\infty} J^{-\frac12}\varphi_0^{(LT)}=\varphi_0^{(SC)}.
$$
One can also see this connection more clearly by taking a second look
at the discrete dynamical system (3.9).
In order to clarify this we make a symplectic scaling tranformation
of $\kappa$ to get a new dynamical system
$$
\tau(x,\xi;J) =
(\kappa_x(x,J^\frac12\xi;J),J^{-\frac12}\kappa_\xi(x,J^\frac12\xi;J)).\tag{5.1}
$$
This system then have its incoming manifold described by the graph
of $J^{-\frac12}\varphi_0^{\prime(LT)}$ and $\tau=(\tau_x,\tau_\xi)$ can be
written using the Taylor expansion as
$$
\tau(x,\xi;J)=\pmatrix x\\ \xi\endpmatrix
-J^{-\frac12}\pmatrix \xi \\ V^\prime(x) \endpmatrix
+\frac12 J^{-1} \pmatrix V^\prime(x) \\ V^\dprime(x)\xi \endpmatrix
+J^{-\frac32}\pmatrix 0 \\ r_J(x,\xi) \endpmatrix,\tag{5.2}
$$
where the remainder
$$
r_J(x,\xi) = \frac14V^\dprime(x) V^\prime(x)
+\frac14\int_0^1 \la\nabla^3 V(z_t),(\xi-\frac12 J^{-\frac12}V^\prime(x))
\otimes (\xi-\frac12 J^{-\frac12}V^\prime(x))\ra dt
$$
and $z_t = t x+(1-t)(x-\tau_x)$. The remainder is clearly bounded
uniformly in large $J$.
Applying $\tau$ once, in the limit of large $J$, corresponds to
taking an infinitesimal step along the Hamiltonian flow (4.4)
for the Hamiltonian
(4.3) and in fact by iterating $\tau$ one recovers the Euler-Cauchy
method (up to the remainder term) for integrating the differential equation
(4.4). Thus it is no miracle that one can obtain the
strong-coupling phase from the low-temperature phase and we will in fact adopt
some ideas from theoretical numerical analysis to achieve this end (see [D]).
This seems to be a new way of constructing solutions to the eikonal
equation in a non-local way. Notice that the proof of the
local stable manifold Theorem for continuous dynamical system goes through
the discrete case (see [I]) and in some sense this is also what is done here.
What is more important is probably that as a byproduct we can get
non-local estimates on solutions to the eikonal equation
as strong-coupling limits of corresponding estimates on the $0$'th order
low-temperature phase, which in some sense is easier to handle.
\subhead{Section I.6: Higher order phases}\endsubhead
In this section we return to the equations (3.6) and (4.1) and
obtain equations for higher order phases (in the low-temperature
limit). See [He4] for a more complete exposition. From the discussion
in Sections 3 and 4 we know how to produce $0$'th order phases.
First of all we get an equation
for the coordinate change to $0$'th order now that we now the $0$'th order
phase (see (3.7))
$$
|\nabla_y f_0(x,y)|^2 = W_x(y)=\varphi_0(y)-\varphi_0(x)
+\frac12(V(y)+V(x))+\frac{J}2|x-y|^2.\tag{6.1}
$$
This is an eikonal equation for a convex potential with critical point
at $y=\Phi_J(x)$ and it can be solved (globally)
using the connection with the
$0$'th order phase in the strong-coupling limit.
Here $x$ enters as a parameter in the potential $W_x$.
Before we proceed we need a lemma which appeared in
[Sj1], [He1], [He4] and [HeRa]. It is used
to determine which terms of $\ln\det\nabla^2_y f(x,y;h)$ one should use
at a given order.
\proclaim{Lemma I.6.1} Let $h\rightarrow M(h)$ be a smooth family of positive
$m\times m$ matrices with an asymptotic expansion
$M(h)\sim \sum_{k\geq 0}M_kh^k$ and suppose $M_0>0$.
Then $L(h)=\ln\det M(h)$ has an asymptotic expansion
$L(h)\sim\sum_{k\geq 0}L_kh^k$ where
$$
L_0 = \ln\det M_0
$$
and for $k\geq 1$
$$
L_k = \sum_{n=1}^{k}\sum\Sb{j_1,\dots,j_n\geq 1}\\{j_1+\cdots+ j_n = k}
\endSb (-1)^{n-1}(n-1)!\tr\{\Pi_{i=1}^n M_0^{-1}M_{j_i}\}.
$$
\endproclaim
As for the $1$'st order phase we get
$$
\varphi_1(y) - \varphi_1(x) = 2\nabla_y f_0(x,y)\cdot\nabla_y f_1(x,y)
- \ln\det\nabla^2_yf_0(x,y) + F_0.\tag{6.2}
$$
Considering this equation along the critical manifold $y=\Phi_J(x)$ gives
$$
\varphi_1(x) = \varphi_1(\Phi_J(x)) + \ln\det\nabla_y^2 f_0(x,\Phi_J(x)) - F_0.
\tag{6.3}
$$
The idea is to choose
$$
F_0 = \ln \det\nabla^2_y f_0(0,0)
$$
as the eigenvalue to $0$'th order and then iterate keeping
in mind that $\Phi_J^{(k)}(x)\rightarrow 0$ as $k\rightarrow\infty$.
Subtracting the constant part of $\ln\det(\nabla_y^2f_0)(x,\Phi_J(x))$ will
ensure convergence. In order to control the regularity of this construction
we will need to have good estimates of iterates of $\Phi_J$ and
its derivatives.
The coordinate change to $1$'st order satisfies a standard transport
equation
$$
2\nabla_y f_1(x,y)\cdot\nabla f_0(x,y) = \varphi_1(y)-\varphi_1(x)+\ln\det
\nabla_y^2f_0(x,y) -F_0\tag{6.4}
$$
which can be solved explicitly around $y=\Phi_J(x)$. In particular
it can be solved in the same domains for which we construct the $0$'th order
coordinate change. (See for example [He1], [He4] or [Sj1]).
The rest of the way is simply a repetition of the procedure described
for the $1$'st order phase and coordinate change.
First set the eigenvalue at $k$'th order equal to the known
term in the $k$'th order equation, obtained from (3.6),
evaluated at $(0,0)$
and iterate towards $(0,0)$ along the critical manifold to obtain the $k$'th
order phase. Finally solve the corresponding transport equation and
obtain the coordinate change to $k$'th order.
\vskip1cm
\head{Part II: Constructing the $0$'th order phases}\endhead
\subhead{Section II.1: A family of discrete dynamical systems}\endsubhead
In this section we construct the $0$'th order WKB-phase in the
low-temperature limit.
Let $V\in C^\infty(\Bbb R^m)$ have a non-degenerate local minimum at
$x=0$ with $V(0)=0$. Suppose $V$ is convex in a neigbourhood of $0$ and
define
$$
\Omega = \{x\in\Bbb R^m: V^\dprime(x)>0\}.
$$
Let $D_\infty\subset\Omega$ be an open set containing $0$ and let
$d\in C^\infty(D_\infty)$ be a smooth non-negative
convex function with $d(0)=d^\prime(0)=0$.
For $R>0$ we consider the sets
$$
D_R= d^{-1}([0,R]).
$$
By assumption $D_R$ is convex and closed for small $R$ but this need not be the
case for large $R$ unless for example $\Omega = \Bbb R^m$.
Let
$$
R_0 = \sup \{R>0: D_R\text{ is convex and closed, and }
V^\prime(x)\cdot d^\prime(x)>0, x\in D_R\nnul\}.\tag{1.1}
$$
We can choose $d$ such that $R_0=R_0(d)>0$ since both $d=V_{|\Omega}$
and $d=x^2_{|\Omega}$ are valid choices. For $R0$,
we conclude that the infimum, $\Phi(x)$,
is attained at a point in the
interior of $D_{d(x)}\subset D_\infty$.
In particular this implies that $\Phi$ satisfies the critical equation
$$
J(x-\Phi(x)) = \frac12 V^\prime(\Phi(x)) + \varphi^\prime(\Phi(x))\tag{1.3}
$$
and the inequality
$$
d(\Phi(x))0$. Solving the equation $\rho_t(s)=s$ we find the fixed point
$$
s_f(t) = \sqrt{Jt+\frac14 t^2}.
$$
The estimate (1.15) now implies that the set
$$
\Cal S_{a_l,a_u} = \{\xi\in\Cal S: a_l(x)I\leq\xi^\prime(x)\leq a_u(x)I,x\in D_\infty\}\tag{1.16}
$$
is invariant under $T$ and since
$$
a_l(0)x\in\Cal S_{a_l,a_u},
$$
we find that it is a non-empty subset of $\Cal S$.
%and one easily verifies that $\rho_t$ is a strict contraction on $[0,\infty)$,
%$s_f(t) < \rho_t(s) < s$ for $s> s_f(t)$ and $s<\rho_t(s)0$ such that
$$
\sup_{(x,\xi,t)\in K} |\tilde{\Psi}_t^n(x,\xi)-\Psi_t(x,\xi)|\leq C_K n^{-2}.
$$
\endproclaim
\proof
Let $K\subset \Cal P$ be compact. We can assume that if $(x,\xi,t)\in K$,
$t\geq 0$ ($t\leq 0$), then $(\psi_{t-s}(x,\xi),r)\in K$ for $0\leq r\leq s\leq t$
($0\geq r\geq s\geq t$).
Using Taylor's formula on the Hamiltonian flow around $t=0$ we find
$$
\Psi_t(x,\xi) = \pmatrix x\\ \xi\endpmatrix
-t\pmatrix \xi \\ V^\prime(x)\endpmatrix + \frac{t^2}2 \pmatrix
V^\prime(x) \\ V^\dprime(x)\xi \endpmatrix + t^3 R_t(x,\xi),
$$
where the remainder $R_t(x,\xi)$ is bounded uniformly in $K$.
By this computation we easily conclude that
the error after one iteration is
$$
\sup_{(x,\xi,t)\in K}|\Psi_{\frac{t}{n}}(x,\xi)-\tilde{\tau}(x,\xi;t,n)|
\leq C_0 n^{-3}.
$$
The next step will be to estimate how much $\tilde{\tau}$ distorts phase-space.
We compute for $(x,\xi)$ and $(y,\eta)$ in $\Bbb R^m\times\Bbb R^m$.
Using (I.5.2) and (2.2) we find the estimate
$$
|\tilde{\tau}(x,\xi;t,n)-\tilde{\tau}(y,\eta;t,n)| \leq \left(1 +
\frac{C_1}{n}\right)\left|\pmatrix x\\\xi\endpmatrix -\pmatrix y\\\eta
\endpmatrix\right|
+\frac{C_2}{n^3},\tag{2.4}
$$
where the constants can be chosen locally uniformly in $(x,\xi,y,\eta,t)$
and $n\geq 1$. Pick $C_1$ and $C_2$ such that the estimate holds uniformly
in $(x,\xi,y,\eta,t,n)$ with $(x,\xi,t)\in K$, $(y,\eta,t)\in K$ and $n\geq 1$.
We can now make the estimate
$$
\align
|\Psi_t(x,\xi)-\tilde{\Psi}^n_t(x,\xi)|&\leq
|\Psi_t(x,\xi)-\tilde{\tau}(\Psi_{t-\frac{t}{n}}(x,\xi);t,n)|\\
&\qquad +|\tilde{\tau}(\Psi_{t-\frac{t}{n}}(x,\xi);t,n) -
\tilde{\tau}^{(n)}(x,\xi;t,n)|\\
&\leq \left(1 + \frac{C_1}{n}\right)|\Psi_{t-\frac{t}{n}}(x,\xi) -
\tilde{\tau}^{(n-1)}(x,\xi;t,n)| + \frac{C_0 + C_2}{n^3}.
\endalign
$$
Iterating this estimate using the choice of $C_1$ and $C_2$
and the inequality
$$
\left(1+\frac{C_1}{n}\right)^k\leq \left(1+\frac{C_1}{n}\right)^n,
\mfor 0\leq k\leq n,
$$
gives
$$
|\Psi_t(x,\xi)-\tilde{\Psi}^n_t(x,\xi)|\leq
\frac{C_0 + C_2}{n^2}\left(1+\frac{C_1}{n}\right)^n.\tag{2.5}
$$
Since $(1+\frac{C_1}{n})^n\rightarrow e^{C_1}$ as $n\rightarrow\infty$,
we conclude the result.
\endproof
Let
$$
\Cal P_- = \{(x,\xi,t)\in\Cal P : t\leq 0\}.
$$
Then we have
\proclaim{Corollary II.2.2} The family of diffeomorphisms $\Psi_t^n$, $t\leq 0$
converges to the Hamiltonian flow (I.4.4), for the Hamiltonian (I.4.3),
locally uniformly in $\Cal P_-$.
In particular for any compact $K\subset \Cal P_-$ there exists $C_K>0$ such that
$$
\sup_{(x,\xi,t)\in K} |\Psi_t(x,\xi)-\Psi_t^n(x,\xi)|\leq C_Kn^{-2}.
$$
\endproclaim
\subhead{Section II.3: Solving the eikonal equation}\endsubhead
In this section we return to the potentials considered in Section II.1
and the family of fixed points $\eta_J = J^{-\frac12}\xi_J$
connected to the rescaled
diffeomorphism $\tau$ (see (I.5.1)). We know that
$$
b_l(x)I\leq \eta_J^\prime(x)\leq b_u(x)I,
$$
where
$$
b_l(x) = J^{-\frac12}a_l(x)\mand b_u(x) = J^{-\frac12}a_u(x).
$$
Consider an $00$. There exist
$N_0>0$ and $C>0$ such that
$$
|\Psi_t^n(x,\eta)-\Psi_t^n(x,\eta^\prime)| < \epsilon + C|\eta-\eta^\prime|
$$
uniformly in $n\geq N_0$ and $\eta,\eta^\prime\in\Sigma_R$.
\endproclaim
\proof
We estimate as for (2.4)
$$
\align
|\tau(x,\eta;\frac{n^2}{t^2})&-\tau(y,\eta^\prime;\frac{n^2}{t^2})|\\
&\leq \left(1 + \frac{C_1 |t|}{n} + \frac{C_2t^2}{n^2}\right)
\left|\pmatrix x\\ \eta\endpmatrix
-\pmatrix y \\ \eta^\prime\endpmatrix \right| + \frac{C_3 |t|^3}{n^3},
\endalign
$$
where the constants are chosen uniformly in $D_R\times \Sigma_R$.
By iterating this estimate we get (see the argument for (2.5))
$$
|\Psi_t^n(x,\eta) -\Psi_t^n(x,\eta^\prime)|\leq \left(1 + \frac{C_1
|t|}{n}+\frac{C_2t^2}{n^2}\right)^n\left\{\frac{C_3 |t|^3}{n^2} +
|\eta-\eta^\prime|\right\}
$$
and the lemma follows.
\endproof
The following lemma is the first example of a statement which is proved
by choosing invariant subsets of $\Cal S_{a_l,a_u}$ (see (1.16))
thereby implying
statements for the fixed point. This idea will be central for the next section.
\proclaim{Lemma II.3.2} The map $J\rightarrow\eta_J$ is smooth and we have the
following bound for all $00$.
\endproclaim
\proof
That $\xi_J$ (and hence $\eta_J$) is smooth with respect to $J>0$
follows from the stable manifold
Theorem, see [I]. It can be verified independently however by
applying the idea of this proof to higher order derivatives in $J$.
This means that $T$ can be viewed as a map on the set
$\Cal S^1=C^\infty((0,\infty);\Cal S_{a_l,a_u})$ and it has a unique
fixed point $\xi_J$.
In analogy with the proof of Theorem 1.1 we write for $\xi\in\Cal S^1$
with associated vector-field $\Phi$
$$
\sigma_\xi(J;x) = \sup_{y\in D_{d(x)}} |\frac{d}{dJ}\xi(J;y)|.
$$
Taking the derivative with respect to $J$ of the critical relation in (1.7),
we get
$$
\frac{d}{dJ}\Phi = J^{-1}\Phi^\prime(x-\Phi-\frac{d}{dJ}\xi(\Phi;J)).
$$
This implies
$$
\frac{d}{dJ}T\xi = (I-\Phi^\prime)(x-\Phi) + \Phi^\prime\frac{d}{dJ}
\xi(\Phi;J)
$$
which together with (1.4), (1.8) and (1.20) gives the estimate
$$
\sigma_{T\xi}(J;x)\leq t_J(x)+\theta(x)\sigma_\xi(J;x),
$$
where
$$
\theta = \frac{J}{J+\frac12\lambda_l+a_l}
$$
and
$$
t_J(x)=\left(\frac{\frac12\lambda_u(x)+a_u(x)}{J+\frac12\lambda_u(x)+a_u(x)}
\right)^2 \sup_{y\in D_{d(x)}}\|y\|.
$$
As in the proof of Theorem 1.1, we can consider the (contraction) map
$\rho_{t,\theta}(s) = t +\theta s$ and conclude that the fixed point $\xi_J$
must satisfy the estimate $\sigma_{\xi_J}\leq s_f(t_J,\theta)$, where
$$
s_f(t_J,\theta) = \frac{t_J}{1-\theta}
$$
is the fixed point of $\rho_{t_J,\theta}$.
Note that for $J$ large we have by (1.13)
$$
t_J\sim J^{-1}\lambda_u\sup_{y\in D_{d(x)}}\|y\|\mand
1-\theta\sim J^{-\frac12}\sqrt{\lambda_l}
$$
uniformly in $D_R$, $0t_c>-\infty$.
Let $\epsilon>0$.
By Corollary 2.2 there exists $N_1>0$ such that
$$
|\Psi_t(x,\xi)-\Psi^n_t(x,\xi)|\leq \frac{\epsilon}3,
$$
for $n\geq N_1$.
By Lemma 3.1, applied with $(\eta,\eta^\prime) = (\xi,\eta_{J_k})$,
we thus get an $N_0$ and a $K_0$ such that
$$
|\Psi_t(x,\xi)-\Psi_t^n(x,\eta_{J_k}(x))|\leq \frac{2\epsilon}3,\tag{3.1}
$$
for $n\geq \max\{N_0,N_1\}$ and $k\geq K_0$.
Let $\tilde{J}_k = \frac{([t\sqrt{J_k}]+1)^2}{t^2}$, $t<0$. Then $n_{k,t}=
t\sqrt{\tilde{J}_k}$ is an integer and
$$
0\leq \tilde{J}_k- J_k\leq \frac{4}{t^2}(1+t\sqrt{J_k}).
$$
Using Lemma 3.2 we thus get (uniformly in $k$ and $D_{d(x)}$)
$$
\align
|\eta_{J_k}-\eta_{\tilde{J}_k}|&\leq \int_{\tilde{J}_k}^{J_k}|
\frac{d}{dJ}\eta_J|dJ\\
&\leq C\frac{1+t\sqrt{J_k}}{t^2 J_k}.
\endalign
$$
This estimate shows,
in conjunction with (3.1), that for each $0> t > t_c$ there exists
$K_0>0$ large enough such that
$$
|\Psi_t(x,\xi)-\Psi_t^{n_{k,t}}(x,\eta_{\tilde{J}_k}(x))|\leq\epsilon,
$$
for $k\geq K_0$. Here we have used Lemma 3.1 with
$(\eta,\eta^\prime)=(\eta_{J_k},\eta_{\tilde{J}_k})$.
This implies that $\Psi_t(x,\xi)$ can be approximated by
elements of $D_{d(x)}\times\Sigma_{d(x)}$, see (1.17). Since this
set is closed,
$\Psi_t(x,\xi)$ must be there itself. We have now shown
$$
\Psi_t(x,\xi)\in D_{d(x)}\times \Sigma_{d(x)},\mfor 0\geq t > t_c.
$$
The flow is thus contained in a compact set and we find by general
theory that it can be extended beyond $t_c$, see [HiSm]. This concludes
the proof.
\endproof
\proclaim{Corollary II.3.4} The accumulation point $(x,\xi)$ lies on the
outgoing manifold for the hyperbolic fixed point $(0,0)$ of the
Hamiltonian flow.
\endproclaim
\proof
Let $x\in D_\infty\nnul$. We write $(x(t),\xi(t))$ for the
flow $\Psi_t(x,\xi)$ with $\Psi_0(x,\xi)=(x,\xi)$.
Suppose there exist $\epsilon>0$ and a sequence $\{t_n\}_{n\in\Bbb N}$
with $t_n\rightarrow -\infty$ such that
$$
x(t_n)^2+\xi(t_n)^2\geq\epsilon.
$$
First notice that by Hamiltons equations (I.4.4) and convexity of $V$
the quantities
$$
h_1(t)=x(t)\cdot\xi(t)\mand h_2(t)=\xi(t)\cdot V^\prime(x(t))
$$
are increasing in $t$ (and positive at $t=0$).
Now compute for $t,t_0<0$
$$
\align
x(t)^2+\xi(t)^2
&= \int_{t_0}^t\int_{t_0}^s \frac{d}{dr}(h_1(r)+h_2(r))drds\\
&\qquad + (t-t_0)(h_1(t_0)+h_2(t_0)) +x(t_0)^2+\xi(t_0)^2\\
&\geq -|t_0-t|(h_1(0)+h_2(0)) + x(t_0)^2+\xi(t_0)^2.
\endalign
$$
This shows that for $\delta = \frac{\epsilon}{2(h_1(0)+h_2(0))}$ we have
$$
x^2(t)+\xi^2(t)\geq\frac{\epsilon}2\mfor |t_n-t|\leq\delta, n\in\Bbb N.
$$
In particular this implies the bound (for $\rho = \min\{1,\lambda_l(x)\}$)
$$
\align
h_1(t_n) &= - \int_{t_n}^0 \xi(s)^2 + x(s)\cdot V^\prime(x(s))ds + h_1(0)\\
&\leq -\rho\int_{t_n}^0 x(s)^2+\xi(s)^2ds + h_1(0)\\
&\leq -n\rho\epsilon\delta + h_1(0),
\endalign
$$
which contradicts the requirement given by Lemma 3.3 that the trajectory
stays in the compact set $D_{d(x)}\times\Sigma_{d(x)}$.
\endproof
This result shows that the gradient of the $0$'th order low-temperature phase
$\varphi_0$ converges to the gradient of a solution to the eikonal
equation. More precisely
\proclaim{Theorem II.3.5} Let $\varphi_0$ be as in Theorem II.1.1. The limit
$$
\lim_{J\rightarrow\infty} J^{-\frac12}\varphi_0 = \psi
$$
exists and is attained locally uniformly in $D_\infty$.
The function $\psi$ is a smooth non-negative convex function with
$\psi(0)=0$ and it satisfies the eikonal equation
(I.4.2) in $D_\infty$. We furthermore have the bounds
$$
\sqrt{\lambda_l(x)}I\leq \psi^\dprime(x)\leq \sqrt{\lambda_u(x)}I,\mfor x
\in D_\infty.
$$
\endproclaim
\proof
As mentioned earlier the outgoing manifold may have many branches in
$D_\infty\times \Bbb R^m$. We will argue that there is at most one
point $(x,\xi)\in D_\infty\times\Bbb R^m$ on the outgoing manifold
which propagated by $\Psi_t$ does not leave this set on its way to $(0,0)$.
If this is the case then Corollary 3.4 shows that for each $x$ such a
$\xi = \eta_\infty(x)$
exists and since it is unique it is the limit
of $\eta_J(x)$ as $J\rightarrow\infty$ and hence Lipschitz.
Since the outgoing manifold of the Hamiltonian flow is $m$ dimensional
and smooth we conclude that $\eta_\infty$ is smooth and
the branch of the outgoing manifold in $D_\infty\times\Bbb R^m$
going through $(0,0)$ is the graph of $\eta_\infty$. The function $\psi$ in the
theorem is chosen such that $\psi^\prime = \eta_\infty$, which is possible
since $\eta_\infty^\prime$ is symmetric.
That $\psi$ solves the eikonal equation follows from conservation of energy.
Let $x\in D_\infty$ and suppose there exists $\xi_1,\xi_2\in\Bbb R^m$
such that $\Psi_{-t}(x,\xi_1)=(x_1(-t),\xi_1(-t))$ and
$\Psi_{-t}(x,\xi_2)=(x_2(-t),\xi_2(-t))$ stays in $D_{d(x)}\times\Bbb R^m$
and converges to $(0,0)$ as $t\rightarrow +\infty$.
Let $h(t) = (x_1(-t)-x_2(-t))\cdot (\xi_1(-t)-\xi_2(-t))$.
We have
$$
\dot{h}(t) = -|\xi_1(-t)-\xi_2(-t)|^2-(x_1(-t)-x_2(-1))\cdot
(V^\prime(x_1(-t))-V^\prime(x_2(-t))).\tag{3.4}
$$
Since $x_1(-t),x_2(-t)\in D_{d(x)}$ and $V$ is convex on the convex
set $D_{d(x)}$ with Hessian bounded from below, we find by Taylor's formula
$$
\dot{h}(t) \leq - |\xi_1(-t)-\xi_2(-t)|^2-\rho|x_1(-t)-x_2(-t)|^2,
$$
for some $\rho>0$.
This shows on one hand that $h(t)$ is non-increasing as $t\rightarrow +\infty$.
On the other hand we know that $h(0)=0$ and
$$
\lim_{t\rightarrow\infty} h(t)=0
$$
by assumption. Hence $h(t)=0$ for any $t\geq 0$.
Since the right-hand side of (3.4) is now forced to vanish for all
$t\geq 0$, we have in particular that $\xi_1=\xi_2$. This concludes the proof.
\endproof
We note that the estimates provided by $a_l$ and $a_u$ are not optimal
since they are required to be monotone, see (1.10). Only in the case where
$V$ is of quadratic type do they even capture the growthrate. Around the
minimum however these bounds are optimal.
\subhead{Section II.4: Further results for the $0$'th order phases}\endsubhead
In this section we discuss some additional properties of the constructions
presented in Part II. The first observation concerns symmetries.
Let $G\subset O(m)$ be a subgroup of the orthogonal group.
We say a function $f:\Bbb R^m\rightarrow \Bbb R$ is $G$-invariant if
$$
f(gx) = f(x),\mfor g\in G.
$$
Notice that if $\tilde{d}$ is a comparison function for a
$G$-invariant potential $V$ then
$$
d(x) = \int_G \tilde{d}(gx)dg
$$
is a $G$-invariant comparison function for $V$. Here $dg$ is the
Haar measure on $G$. Now let $d$ be $G$-invariant
(then $g D_R=D_R$, for $g\in G$).
The following result follows by replacing $\tilde{\Cal S}$ and $\Cal S$
in Section 1 by
$$
\tilde{\Cal S}_G =
\{\varphi\in\tilde{\Cal S}: \varphi(gx)=\varphi(x), g\in G, x\in D_\infty\}
$$
and
$$
\Cal S_G = \{\xi\in \Cal S: g^{-1}\xi(gx) = \xi(x),g\in G,x\in D_\infty\},
$$
noting that the construction using functions from these classes
is invariant under $G$.
\proclaim{Proposition II.4.1} Let $G$ be a subgroup of $O(m)$ and suppose
$V$ is $G$-invariant. Then the low-temperature phase
constructed in Theorem II.1.1 and the solution to the eikonal equation
given by Theorem II.3.5 are $G$-invariant.
\endproclaim
The next observation is concerned with extending the domain for which we have
a solution. Let $\Cal D=\Cal D(V)$ denote the class of comparison functions,
$d$, for a potential $V$
in the sense of Section 1. For each $d$ we have a maximal convex
domain $D_\infty(d)$ in which the construction works. Pathching
these domains together we get solutions in domains of the form
$$
\cup_{d\in\Cal D} D_\infty(d).\tag{4.1}
$$
This can be used in applications to get solutions in some non-convex domains
and we will use it in a later paper to construct solutions in $l^\infty$
neighbourhoods of the critical point with control in high dimension.
See [Sj1] and [He4]. In [So] local solutions to the eikonal equation are
constructed in $l^2$ balls with radius scaling like the square root
of the dimension.
Let $d\geq 0$ be a smooth convex function on an open subset of $\Bbb R^m$
containing zero.
Let $D_\infty = \mycirc{D}_R$ for some $R>0$ for which $D_\infty$
is convex. For such $R$ and $\sigma>0$ we consider the potential class
$$
\Cal V_{R,\sigma} = \{V\in C^\infty(D_\infty): V\geq 0,V(0)=0,
V^\dprime\geq\sigma I, V^\prime\cdot d^\prime>0
\text{ in } D_\infty\nnul\}.
$$
Write
$$
\theta(\sigma) = \frac{J}{J+\frac12\sigma+\sqrt{J\sigma+\frac14\sigma^2}}.
$$
\proclaim{Proposition II.4.2} Let $V_1,V_2\in\Cal V_{R,\sigma}$. We have
$$
\sup_{x\in D_\infty}\|\nabla(\varphi_{0,1}-\varphi_{0,2})\|\leq
\frac1{2(1-\theta(\sigma))}\sup_{x\in D_\infty}\|\nabla(V_1-V_2)\|,
$$
and for the associated vector-fields, see (1.7), we have
$$
\sup_{x\in D_\infty}\|\Phi_1-\Phi_2\|\leq \frac{\theta(\sigma)+1}{2J(1-\theta(\sigma))}
\sup_{x\in D_\infty}\|\nabla(V_1-V_2)\|.
$$
As for the solutions to the eikonal equation $\psi_1$ and $\psi_2$ we have
$$
\sup_{x\in D_\infty}\|\nabla(\psi_1-\psi_2)\|\leq \frac1{2\sqrt{\sigma}}
\sup_{x\in D_\infty}\|\nabla(V_1-V_2)\|.
$$
\endproclaim
\proof Write
$$
\Cal S_\sigma = \{\xi\in\Cal S:\xi^\prime\geq\sqrt{J\sigma+\frac14\sigma^2}I\}.
$$
Then $\Cal S_\sigma$ is invariant under $T_1$ and $T_2$ by the choice
of $\Cal V_{R,\sigma}$ and the proof of the lower bound in Theorem 1.1.
Here $T_i$ will denote the map $T$ associated with the potential $V_i$.
Let $\xi_1,\xi_2\in\Cal S_\sigma$ and write $\Phi_1$ and $\Phi_2$ for
the vector-fields obtained from $\xi_1$ and $\xi_2$ using
the critical equation in (1.7) with potentials $V_1$ and $V_2$
respectively.
We compute as for the argument which gave the contraction property
of $T$ on $\Cal S$, see (1.11), and get for $\xi_1,\xi_2\in\Cal S_\sigma$.
$$
\sup_{x\in D_\infty}\|\Phi_1-\Phi_2\|\leq\frac{\theta(\sigma)}{J}
\sup_{x\in D_\infty}\|\xi_1-\xi_2\|.
$$
Choosing $\xi_1$ and $\xi_2$ as the fixed points of $T_1$ and $T_2$ restricted
to $\Cal S_\sigma$ thus imply
$$
\sup_{x\in D_\infty}\|\xi_1-\xi_2\| \leq \frac1{2(1-\theta(\sigma))}
\sup_{x\in D_\infty}\|\nabla(V_1-V_2)\|.
$$
This gives the first estimate since $\nabla\varphi_{0,i}=\xi_i$.
The second estimate now follows directly from the first and the last
follows from Theorem 3.5 by noting that
$$
\lim_{J\to\infty}J^{-\frac12}\frac1{2(1-\theta(\sigma))}=
\frac1{2\sqrt{\sigma}}.
$$
\endproof
This proposition is a special case of a more general result for parameter
dependent potentials $V_z$ which gives continuity of mixed derivatives
of the low-temperature phase with respect to the potential,
provided the potentials are taken from some a priori class
(like $\Cal V_{R,\sigma}$).
We do not elaborate further but refer the reader to a coming paper in which
we will control the constructions presented here with respect to the dimension $m$
using the framework of standard function calculus (see [Sj2]).
\vskip1cm
\head{Part III: Global analysis}\endhead
\subhead{Section III.1: Restriction to global minima}\endsubhead
In this part of the paper we wish to analyze the localization
properties of the transfer operator in the low-temperature limit
and use the WKB-construction of Helffer, as presented in
Sections I.3 and I.6, of the localized first eigenfunction to gain
information on the corresponding eigenvalue.
We note that the methods presented
here does not directly extend to give control with respect to high dimension.
Let $V\in C^\infty(\Bbb R^m)$ be a non-negative potential with a finite
number of non-degenerate global minima, $\{z_1,\dots,z_k\}$, with $V(z_i) = 0$.
Choose $R$ small enough such that $V$ is convex in each of the $k$
disjoint balls
$$
B_i=B(z_i,R) = \{x\in\Bbb R^m:|x-z_i|\leq R\}.
$$
Write $B_0 = \Bbb R^m\backslash \cup_{i=1}^k B_i$ and assume
$$
\rho = \inf_{x\in B_0} V(x)>0.
$$
We write
$$
r_0 = \min_{1\leq i< j\leq k} \dist(B_i,B_j).
$$
Let $K_i$, $1\leq i\leq k$, be the restriction of the full transfer operator
$K$ (see Section I.3) to the ball $B_i$. That is the bounded operator
with the kernel
$$
K_i(x,y) = K(x,y)_{|B_i\times B_i}
$$
on the Hilbert space $L^2(B_i)$.
One can verify that $K$ becomes trace class if $V$ satisfies some growth
estimate at infinity (see [He5]). These properties are however
not needed for the analysis here.
Let $\chi_i$ be the characteristic function for the
ball $B_i$, $0\leq i\leq k$.
First we estimate the contribution from the region away from the
global minima
$$
\|\chi_0 K\| \leq \sup_{x\in B_0} e^{-\frac1{2h}V(x)}=e^{-\frac{\rho}{2h}}.
\tag{1.1}
$$
Let $\varphi_1,\varphi_2\in L^2(\Bbb R^m)$. For $1\leq i < j\leq k$ we have
$$
\align
|\langle \chi_i\varphi_1, K\chi_j\varphi_2\rangle|
&\leq (\pi h)^{-\frac{m}2}\int_{\Bbb R^m}\int_{\Bbb R^m}\chi_i(x)\chi_j(y)
e^{-\frac{J}{2h}|x-y|^2}|\varphi_1(x)||\varphi_2(y)|dxdy\\
&\leq (\pi h)^{-\frac{m}2}|B|e^{-\frac{Jr_0^2}{2h}}\|\varphi_1\|\|\varphi_2\|,
\endalign
$$
where $|B|$ denotes the volume of the balls $B_i$.
This implies the following estimate on the coupling between minima
$$
\|\chi_iK\chi_j\|\leq (\pi h)^{-\frac{m}2}e^{-\frac{Jr_0^2}{2h}}
\mfor 1\leq i 0$.
We choose the first eigenfunction $\psi_1$ normalized and positive and
we write $\lambda_1^i$, $1\leq i\leq k$, for the first eigenvalue of the
operator $K_i$. Let
$$
\mu_1= \max_{1\leq i\leq k} \lambda_1^i.
$$
The estimates (1.1-2) show on one hand that
$$
\align
\lambda_1 &\leq \sum_{i=1}^k \la \chi_i\psi_1,K\chi_i\psi_1\ra
+Ce^{-\frac{\delta}{h}}\\
&\leq \sum_{i=1}^k \lambda_1^i\|\chi_i\psi_1\|^2 + Ce^{-\frac{\delta}{h}}\\
&\leq \mu_1 + Ce^{-\frac{\delta}{h}}.\tag{1.3}
\endalign
$$
On the other hand we write $\psi_1^i$ for the normalized first eigenfunction
of the operator $K_i$ extended to the whole space by the $0$ function.
Again we choose it positive.
Then by the variational principle
$$
\align
\lambda_1&\geq \max_{1\leq i\leq k}\la\psi_1^i,K\psi_1^i\ra\\
&= \max_{1\leq i\leq k}\la\psi_1^i,K_i\psi_1^i\ra\\
&=\mu_1.\tag{1.4}
\endalign
$$
We have the a priori estimate (see [He2] and [He5])
$$
00$.
The lower bound follows from an application of Segal's Lemma (or harmonic
approximation which also applies to the $\mu_1^i$'s).
The estimates (1.3-5) give
\proclaim{Theorem III.1.1} There exists $h_0>0$ such that
$$
\mu_1\leq\lambda_1\leq \mu_1(1+Ce^{-\frac{\delta}{h}})
$$
for $00$ such that
$$
0\leq \ln\frac{\lambda_1}{\lambda_2} \leq Ce^{-\frac{\delta}{h}},
$$
for $00$
such that
$$
\ln\lambda_1^i = -F_N^i(h) + O(h^{N+1})
$$
for $00$ such
that $F_N^i(h)>F_N^j(h)$ for $i\in\Cal F^N$, $j\not\in\Cal F^N$ and
$00$ such that
$$
\ln\lambda_1 = - F^i_N(h) + O(h^{N+1})
$$
for $i\in\Cal F^N$ and $00$ such that
for $00$ and $N\in\Bbb N$,
and this together with Theorem 1.1 proves the result.
\endproof
In the proof of the next result we will need another
version of the $\mu_2$ used in the proof of the previous result.
Namely
$$
\mu_2^\dprime = \max_{i\in\Cal F^\infty}\{\lambda_2^i\},
$$
where the $\lambda_2^i$'s are the second eigenvalues of the restricted
operators $K_i$ (see Theorem 2.5). One can use harmonic approximation to
verify that $\lambda_1^i-\lambda_2^i\geq C>0$ uniformly in small $h$,
see [He5]. Hence we have
$$
\mu_1 - \mu_2^\dprime\geq C>0.\tag{2.1}
$$
\proclaim{Proposition III.2.4} Let $h_0>0$ be small enough.
For all $00$ and $\sum_{i\in\Cal F\infty} \omega_i^2 = 1$ such that
$$
\|\psi_1 -\psi_{1,\ul{\omega}}\| \leq Ce^{-\frac{\delta}{h}},
\mwhere
\psi_{1,\ul{\omega}} = \sum_{i\in\Cal F^\infty} \omega_i\psi_1^i.
$$
\endproclaim
\proof
Let, for $i\in\Cal F^\infty$,
$$
\tilde{\omega}_i = \la \psi_1,\psi_1^i\ra\mand \omega_i =
\frac{\tilde{\omega}_i}{\sqrt{\sum_{i\in\Cal F^\infty}\tilde{\omega}_i^2}}.
$$
We introduce the subspace $\Cal H_0\subset L^2(\Bbb R^m)$
of linear combinations of the functions
$\psi_1^i$, $i\in\Cal F^\infty$ (extended to $0$ outside $B_i$).
Notice that $\varphi\in \Cal H_0^\perp$ will satisfy
that $\varphi_{|B_i}\perp\psi_1^i$, for $i\in\Cal F^\infty$.
We wish to prove that
$$
|\la\varphi,\psi_1\ra|\leq Ce^{-\frac{\delta}{h}}\tag{2.2}
$$
for $\varphi\in\Cal H_0^\perp$. This will imply the result since
$\psi_{1,\ul{\omega}}$ is exactly the orthogonal projection of $\psi_1$
onto $\Cal H_0$.
The estimate (2.2) holds by Corollary 2.3 for $\varphi$ with support
outside
$$
B_\infty = \cup_{i\in\Cal F^\infty} B_i.
$$
As for functions with support inside $B_\infty$ we introduce the
projection onto $\Cal H_0$
$$
P\varphi = \sum_{i\in\Cal F^\infty}\la\psi_1^i,\varphi\ra\psi_1^i
$$
and the operator (on $L^2(B_\infty)$)
$$
K_\infty = \sum_{i\in\Cal F^\infty} K_i.
$$
Notice that $PK_\infty = K_\infty P$ and $I-P$ is the projection onto
functions in $\Cal H_0^\perp$ restricted to $B_\infty$.
We can now estimate using $K_{\infty|\range (I-P)}\leq\mu_2^\dprime$
$$
\align
\|(I-P)\chi_\infty\psi_1\|^2 &=
\la (I-P)(\mu_1-K_\infty)^{-1}(I-P)\chi_\infty\psi_1,
(\mu_1-K_\infty)\chi_\infty\psi_1\ra\\
&\leq (\mu_1-\mu_2^\dprime)^{-1}\|(I-P)\chi_\infty\psi_1\|
\|(\mu_1-K_\infty)\chi_\infty\psi_1\|.\tag{2.3}
\endalign
$$
By (1.2) we have
$$
\|K_\infty-\chi_\infty K\chi_\infty\|\leq Ce^{-\frac{\delta}{h}},
$$
which we combine with (2.1) and (2.3) to get the estimate
$$
\align
\|(I-P)\chi_\infty\psi_1\|&\leq C\|(\mu_1-K_\infty)\chi_\infty\psi_1\|\\
&\leq C\|(\mu_1 - K)\chi_\infty\psi_1\|+Ce^{-\frac{\delta}{h}}\\
&\leq C\|(\mu_1-K)\psi_1\| + Ce^{-\frac{\delta}{h}},
\endalign
$$
where we used Corollary 2.3 in the last step. The result now follows
from an application of Theorem 1.1.
\endproof
If at least two of the weights $\omega_i$ vanishes at most
polynomially in $h$ then the splitting between the
two first eigenvalues of the transfer operator is $O(h^\infty)$.
We omit the proof which follows from Proposition 2.4. See also
Proposition 1.2.
\proclaim{Theorem III.2.5} Suppose $|\Cal F^\infty| = 1$. There exists
$h_0>0$ such that for $00$ uniformly in small $h>0$ implies the result.
One can use harmonic approximation to show
the a priori estimate for $\mu_2^{i_0}$, which is a one well problem
(see [He5]), and the estimate $\mu_1^i\geq C$ is covered by (1.5) (see Theorem
1.1).
\endproof
\subhead{Section III.3: Globally convex potentials}\endsubhead
In the case where the potential $V$ is globally convex
the constructions given in this paper are global as well.
In particular the $0$'th order phases (in the low-temperature
and strong-coupling limits) are also globally convex and if the
Hessian of $V$ is bounded uniformly from below or above then so
are the $0$'th order phases (see Theorems II.1.1 and II.3.5).
In the special case where the Hessian
of the potential is bounded uniformly from both above and below
then one can estimate all the higher order phases globally as well
in the sense that they all grow at most linearly at infinity
and have uniformly bounded derivatives (provided all derivatives of
the potential of order $3$ and higher are uniformly bounded).
We will not prove this result here for two reasons.
The proof is rather technical and lengthy and we will need
and extended version of the proof in a later paper in order to
show that the WKB-constructions considered here are stable
with respect to the standard function class introduced
by Sj{\"o}strand (see [Sj2]). Notice that this result
underscores that the choice of ansatz for the first
eigenfunction employed here is a natural one. We note that one
can use such a result to give an alternative proof of Theorem 2.2.
In the case of the strong coupling limit it seems to be necessary
to know something about the decay of the first eigenfunction
outside wells in order to make a localization argument work.
Alternatively one can make the assumption of uniform bounds
(from above and below) on the Hessian of $V$ and prove the analogue
of Theorem 2.2 in the strong coupling limit using the
uniform control of higher order phases.
It is natural to ask wether or not the $0$'th order phases
can be extended smoothly beyond a region of convexity into
a region of attraction where
$$
x\cdot V^\prime>0,\mfor x\neq 0.
$$
In one dimension the eikonal equation (the $0$'th order phase
in the strong-coupling limit) has the smooth solution
$$
\varphi(x) = \sign(x)\int_0^x \sqrt{V(y)}dy.
$$
However a numerical test for the one-dimensional potential
$$
V(x) = x^2 + \frac34 x\sin(x)
$$
shows that the incoming manifold for the corresponding diffeomorphism
$\kappa$ ceases to be a graph (over configuration space) outside
an interval which grows with $J$ (to include more and more regions of
non-convexity). The $0$'th order low-temperature phase is therefore
not a globally smooth function in this example.
\subhead{Section III.4: Chains of continuous spins}\endsubhead
In this section we will briefly review the consequences of the
results of Part III in the context of a ferromagnetic, $J>0$,
system of continuous spins
arranged on the one-dimensional lattice $\Bbb Z$.
Let $V\in C^\infty(\Bbb R^n)$ be a self-energy. It should
satisfy the conditions introduced in Section 1 and we suppose
for simplicity that it grows at least linearly at infinity.
The energy of a spin-distribution at finite volume
$$
\Lambda =\{-L,\dots,L\}\subset\Bbb Z,\quad L\in\Bbb N,
$$
is given by
$$
H_\Lambda(\ul{\sigma}) = \sum_{i\in\Lambda}V(\sigma_i)
+ \frac{J}2\sum_{i\sim j}|\sigma_i-\sigma_j|^2,
$$
where $i\sim j$ means nearest neighbour with the periodic
boundary condition $-L\sim L$.
The spin distribution $\ul{\sigma}$ takes values in the one-particle space
$\Bbb R^n$. We note that the quantities discussed here are independent of the
choice of boundary condition.
The partition function and free energy of the system is
$$
Z_\Lambda(\beta) = \int_{\Bbb R^{n|\Lambda|}}e^{-\beta
H_\Lambda(\ul{\sigma})}d^{n|\Lambda|}\ul{\sigma}
$$
and
$$
F_\Lambda(\beta) = -\frac{\ln Z_\Lambda(\beta)}{\beta}.
$$
It is well known that in the thermodynamic limit ($L\to\infty$) we have
$$
-\beta\lim_{L\to\infty} F_\Lambda(\beta) =
\frac{n}2\ln\pi - \frac{n}2\ln\beta +\lambda_1(\beta),
$$
where $\lambda_1$ is the highest eigenvalue of the transfer operator
with potential $V$ and coupling constant $J$. In other words
Theorem 2.2 gives a low-temperature
expansion of the free energy in the thermodynamic limit.
Other quantities of interest are expectations of the Gibbs measure,
which at finite volume is given by
$$
d\mu_{\beta,\Lambda} = Z_\Lambda(\beta)^{-1}e^{-H_\Lambda}
d^{n|\Lambda|}\ul{\sigma}.
$$
Of particular interest is the two-point correlation function
$$
\cor_{\beta,\Lambda}(i,j) = \int_{\Bbb R^{n|\Lambda|}}
\sigma_i\cdot\sigma_j\quad d\mu_{\beta,\Lambda}.\tag{4.1}
$$
Another well known connection between the transfer operator and the
thermodynamic limit of one-dimensional spin-systems is the
following relation
$$
\lim_{L\to\infty} \ln\cor_{\beta,\Lambda}(i,j)\sim
-\ln\left(\frac{\lambda_1}{\lambda_2}\right) |i-j|,
$$
asymptotically for large $|i-j|$. In other words the inverse correlation
length of the system is given by the logarithm of the
splitting between the two first eigenvalues of the transfer
operator. In the case where the self-energy is a symmetric double well
we see from Proposition 1.2 that the correlation length can become
exponentially
large in the low-temperature limit. In the case where $|\Cal F^\infty|=1$
(in particular in the case of a unique global minimum) one can apply
Theorem 2.5 to localize the problem and compute the asymptotics
of the correlation lenght at low temperatures using for example
harmonic approximation or WKB-analysis. In [BJS] the low-temperature limit
of the correlation function (4.1)
is determined completely, for $n=1$, under assumptions on the
potential which implies a unique critical point. Here we get
the correlation lenght at low-temperatures
under much weaker conditions on $V$.
We refer the reader
to [F], [K], [He4] and [He5] for a more thorough discussion of the connection
between transfer operators and spin-systems.
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\enddocument
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