Ames W.F., Harrel E.M., Herod J.V. (editors). Differential Equations with Applications to Mathematical Physics
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On Schrodinger Equation in
Large Dimension and
Connected Problems in
Statistical Mechanics
Bernard Helffer
Ecole
Normale
Suphrieure de Paris
DMI-ENS,
45
rue d'Ulm
F75230
Paris C6dex
Abstract
In this article we announce three results concerning semi-classical
techniques in statistical mechanics. The two first results concern
the Schrodinger equation and are obtained in collaboration with
J.
Sjostrand. The last one is
a
stationary phase theorem and can
be considered as
an adaptation
of
a
result
of
J.
Sjostrand in
a
differ-
ent context.
1
Introduction
If
is
a
suitable family
of
C"
potentials on
Ern
parametrized by
m,
there appears to be three connected problems related to the prop-
erties
of
the thermodynamic limit in different contexts
of
statistical
mechanics.
(I) Study the asymptotic behavior of the quantity:
Differential Equations
wi~h
Copyright
@
1993
by Academic Press,
Inc.
Applications
to
Mathematical
All
rights
of
reproduction in
any
form
reserved.
Physics
ISBN
0-12-056740-7
153
154
Bernard
Helffer
as
rn
tends to
00
and control this limit with respect to
h
as
h
tends to
+O.
(11) If
pl(rn,h) is the largest eigenvalue
of
the operator:
Kn(h)
=
exp( -V(m)(2)/2) exp(h2A(m))
.
e~p(-V(~)(z)/2),
study the asymptotic behavior
of
the quantity
-
In pl(rn,
h)/m
as
m tends to
00
and control this limit with respect to
h
as
h
tends to
+O.
(1)
(111) If
Al(rn,h)
is the smallest eigenvalue
of
the Schrodinger oper-
ator:
Sm(h)
=
-h2A(")
t
V(m)(z),
(2)
what is the asymptotic behavior
of
the quantity Al(rn,h)/m
as
rn
tends to
00
and control this limit with respect to
h
as
h
tends to
+O?
These three questions are
of
course strongly related.
If
you think
of
a
potential which is invariant by circular permutation
of
the vari-
ables and "near" in
a
suitable sense
of
the harmonic oscillator, all
these questions are well analyzed
for
fixed
rn
as
h
tends to zero.
(I)
can be treated by application
of
the stationary phase theorem,
(111)
corresponds to
a
semiclassical analysis
of
the Schrodinger op-
erator
at
the bottom (see
[8]
and [20]) and the study
of (11)
can be
considered as
a
pseudo-differential extension
of (11)
(see [2]
or
[5]).
In
particular this study gives
for
example that
-
111(p1(m7h))
=
Al(m,h)
t
om(h2).
(3)
One can get better by proving first (using Segd's lemma) (cf
[18])
the universal inequality:
-
lIl(/L1
(my
h))
5
(
h).
(4)
By monotonicity, one observes also (in the strictly convex case) that
if
Vim' is
a
quadratic potential s.t. Vim)
5
V(")
,
then we have also:
-
1n(/Ly(rn7
h))
5
-
ln(pl(rn,
h))
(5)
Schrodinger Equation in Large Dimension
155
where
&(m,
h)
(the largest eigenvalue of
Km(h;
V:”’))
is explicitly
computable. Equations
(4)
and
(5)
give for example (in the case
when the limits exist):
-
m+m
lim ln(pl(m,h))/m
5
m-+m
lim (&(m,h))/m
in the strictly convex case.
we have:
Another link is that, by Golden Thompson inequality (cf
[IS]),
Tr(exp
-(-A
+
W))
5
=
C,
Jexp(-W)dzm.
Tr(exp(-W/2)
-
exp
A
-
exp(-W/2))
The difficult problem is of course
a
good control with respect to
m
as
m
is large. Another interesting (and more difficult) problem
appears in the same context, in the cases (11) and (111):
(IV) Study the liminf and the limsup of
m
-+
p2(m,h)/pl(m,h)
as
m
-+
00
where
p2(m,
h)
is the second eigenvalue of
Km(h).
(V) Study the lim inf and the limsup of
m
-+
(X2(m,
h)
-
Xl(m,
h))
as
m
-+
00
where
Xz(m,h)
is the second eigenvalue of
Sm(h).
The problem (V) corresponds to the well known problem of the study
of the splitting between the two first eigenvalues. Our motivation
comes from the reading
of
a
course of
M.
Kac
([13])
in which he
develops partially heuristical ideas in order to prove the existence of
phase transition by semi-classical techniques. The results we shall
present here correspond to
a
class containing the model potential:
m
m
...
V(m)(z;
v)
=
(1/4)
c
-
c
In COS~((V/~)’/~(XC
+
z~+l)),
with the convention
(z,+1
=
XI).
156
Bernard
Helffer
Let us briefly recall how
M.
Kac arrived at this potential. He
studied the following model (called Model
A
in section
7
in [13])
whose hamiltonian is given by:
EV(N,M)(4
=
-
c
v(P,Q)ap
aQ
(P,Q)€V(N,W
xV(N,W
with
V(N,M)
=
[l,.
. .
,N]x(Z/nZ) in
Z2,
ap
E
{-l,+l},
J
E
lR;,
h
E
lR$,
vp,p
=
0
and
if
P
=
(k,l)
#
Q
=
(a',/').
limit
-$/kT
can be computed as:
He observes that the free energy per spin in the thermodynamic
-$/kT
=
In 2
-
h/2
+
lim
(lnpl(rn, h)/rn)
m-w
where
pl(rn,
h) is the largest eigenvalue of the rn-dimensional integral
operator
K
given by:
Ir'
=
e~p(-Q(~)/2)
-
exp(-h(-A(m)))
.
e~p(-Q(~)/2)
u
=
J/kT.
A
scaling argument
21,
=
h'/2yk permits one to arrive
essentially to the problem posed in (11).
The detailed
proofs
are
or
will be given elsewhere
([2],
[5],
[6], [7],
~91,
POI,
~31, [24]7 ~51).
2
Schrodinger Equation in Large Dimension
Let
us
consider
Schriidinger Equation in Large Dimension
157
with
m
m
V(")(x;
u)
=
(1/4)
c
xi
-
In
~osh((uh/2)'/~(xk
t
xk+i)),
(9)
k=l
k=1
with the convention:
(xm+l
=
XI).
If
v
<
1/4, the potential is convex
(single well) but,
if
u
>
1/4, we are in the situation of
a
double well.
Let
Xj(m,
h;
u)
be the sequence of the eigenvalues of S("); we are
interested by the problems (111) and
(V).
Let us present the results
which were obtained in this case.
Theorem
2.1
(Cf [9],
[as])
For every
v
in
lR+,
the limit
A(h,v)
=
limm4m(A1(m;
h,
v)/m)
exists.
This is not surprising and it
is
proved following the ideas of sta-
tistical mechanics (see [IS]). Let us observe that
b(h7
u)
-
(Ai(m;
h,
v)/m)l
=
hO(l/m)
by easy arguments and that
J.
Sjostrand [25] proves recently an ex-
ponentially rapid convergence to the thermodynamic limit.
Theorem
2.2
(Cf [9],[25])
Ifu
#
1/4,
A(h,u)
=
lim
(Al(m;
h,v)/rn)
m+m
admits
a
complete asymptotic expansion:
A(h,
u)
x
h
Cjm
Aj(u).hj
as
h
tends to
0.
Moreover,
if
we denote the corresponding semiclas-
sical expansions for
X1
(m;
h,
u)/m
by:
Al(m;
h,v)/m
M
h
-
c
Xj(m, .).hi
there exists
Ico(v)
>
0
s.t. for each
j,
there exists
a
constant
Cj(v),
s.t.
IAj(V)
-
Xj(m,v)l
5
Cj(u).exp(-kom).
ko(v)
and
Cj(u)
can
be
chosen locally independent of
u
in
IR+
\
{
1/4}.
20
The study around
u
=
1/4 is not complete (see however [13],
[6]
for partial result for fixed
m).
158
Bernard
Helffer
Theorem
2.3
(Cf
[24],[10])
If
v
<
1/4
then the splitting between the
two first eigenvalues
A2
and
A1
is controlled
by:
h(l-4~)”~
5
Xz(m,h,v)
-
Xl(m,h,v)
I
4Xl(m,h,v)/m.
(10)
The majorization is easily obtained by estimates of the type used
in the proof of
[17]
(see
[9])
and is true for any
v
in
lR+.
The
minorization
([24])
can be obtained by using the maximum Principle
(see
[22])
or
the Brascamp-Lieb inequalities
[l]
as explained in
[lo]
and the strict convexity of the potential is the decisive and unique
assumption.
Theorem
2.4
(Cf
[9])
Let
v
>
1/4
and let us consider
N
the set
in
IN
x
lR+
defined
by
(we write shortly
m
=
O(hWNO))
for
some
C
and
No;
then there
exists
C,,
h,
and
E”
>
0
such that
for
all the
(m,
h)
in
n/
satisfying
0
<
h
5
h,:
mLC.h-No,
(11)
Remark
2.5
Here we observe a very diflerent behavior
in
compari-
son with the case
v
<
1/4
(cf Theorem
2.3)
but we have unfortunately
a restriction on
m.
This
is
probably a technical dificulty. We were
hoping to prove simply that (conjecture given
by
M.
Kac):
lim
(X;z(m,h,v)
-
Xl(m,h,v))
=
0.
rn-oo
This property would have been a
sign
of
a “transition
of
phase”.
The proof of Theorem
2.3
and Theorem
2.4
is based on the fol-
lowing strategy initiated in
[23]
and
[24].
We can distinguish four
steps.
Step
1:
Control in the WKB approximation (just look for approx-
imate eigenfunctions of the type exp(-
f(x,
h)/h))
the dependence
on the dimension
m
as
initiated in
[23]
and
[24].
Of course it is
a
construction which depends only of the germ of the potential
at
the
bottom, but in order to have reasonable estimates we have to assume
Schrodinger Equation in Large Dimension
159
holomorphy in
a
complex open /"-ball. This will give us the formal
expansion of the first eigenvalue.
Step
2:
One compares the
WKB
approximation of the one well
problem and the first eigenvalue of the Dirichlet problem in
a
suf-
ficiently small l"-ball around the point where the minimum
of
the
pot en t
id
was
at
t
ai
lied.
Step
3:
One compares the first eigenvalue
of
the Dirichlet problem
in this small 1"-ball with the first eigenvalue
of
the global problem
in
lR".
In these three steps, one works modulo
m.Olv(hN)
(for any
N)
but the dimension is possibly limited by
m
=
O(h-"J).
Step
4:
One eliminates the restriction on the dimension, because
one controls the rate of convergence in the thermodynamic limit. In
order to analyze the splitting between the two first eigenvalues, let us
recall the following classical formula for the splitting (see for example
[8],
[15],
[20]
and
[21]):
where
7-1
=
y
E
cF*
P(ul,m)2(Z)dx
=
o
{
71
and
~1,"
is the first positive normalized eigenfunction.
The estimates about the splitting are then deduced from
a
ju-
dicious choice
of
p
and of the information
on
the decay
of
Ul,m
in
suitable domains. We observe that, under the assumption
u
>
1/4,
the potential admits two minima and that there exists
6
s.t. the
region
R(6)
defined by:
does not contain these two wells.
&(m,h)
-
xi(m,h)
5
C,.m.h2.(a(m,h,S)2)/(1
-
~(m,h,S)~)
(15)
with
477%
h76)
=
II~l,mllLZ(n(s)).
Theorem
2.4
will be
a
consequence of the following theorem: