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TOWARDS A QUANTUM ANALOG OF WEAK KAM THEORY
Lawrence C. Evans 1
Abstract. We discuss a quantum analogue of Mathers minimization principle for La
grangian dynamics, and provide some formal calculations suggesting the corresponding Euler
Lagrange equation. We then rigorously construct from the dual eigenfunctions of a certain
nonselfadjoint operator a candidate for a minimizer, and recover aspects of weak KAM
theory in the limit as h 0. Regarding our state as a quasimode, we furthermore derivesome error estimates, although it remains a open problem to improve these bounds.
1. Introduction.
This paper proposes an extension of Mathers variational principle [M12, MF] andFathis weak KAM theory [F13] to quantum states. We interpret weak KAM theory tomean the application of nonlinear PDE methods, mostly for firstorder equations, towardsunderstanding the structure of action minimizing measures solving Mathers problem. Asexplained in the introduction to [EG], a goal is interpreting these measures as providinga sort of integrable structure, governed by an associated effective Hamiltonian H, inthe midst of otherwise possibly very chaotic dynamics. The relevant PDE are a nonlineareikonal equation and an associated continuity (or transport) equation.
In this work we attempt to extend this viewpoint and some of the techniques to a quantum setting, in the semiclassical limit. We do so by suggesting an analogue of Mathersaction minimization problem for the Lagrangian L(v, x) = 12 v2W (x), where the potential W is periodic, and formally computing the first and second variations. Thus motivated,we next build a candidate state for a minimizer and discuss at length its properties. Asin the nonquantum case, we come fairly naturally upon an eikonal PDE (with some extraterms) and an exact continuity equation. We next send h 0 and show how the usualstructure of weak KAM theory appears in this limit.
More interesting is understanding if our is a good quasimode, that is, a decent approximate solution of an appropriate eigenvalue problem. This turns out to be so, althoughour error bounds are too weak to allow for any deductions about the spectrum.
1 Supported in part by NSF Grant DMS0070480 and by the Miller Institute for Basic Research in
Science, UC Berkeley
1

Much of the interest in the following calculations centers upon our minimizing, subjectto certain side conditions, the expected value of the Lagrangian, namely the expression
Tn
h2
2D2 W 2dx,
and not the expected value of the Hamiltonian,
Tn
h2
2D2 + W 2dx.
It will turn out that owing to the constraints a minimizer of the former is approximatelya critical point of the later. But the key question, unresolved here, is determining whenthe error terms are of order, say, o(h) in L2.
The calculations below represent improvements upon some ideas developed earlier in[E1]. An interesting recent paper of Anantharaman [A] presents a somewhat similar approach within a probabilistic framework, and Holcman and Kupkas forthcoming paper[HK] is related. Likewise, Gomes [G] found some related constructions for his stochasticanalogue of AubryMather theory. We later discuss also some formal connections withthe stochastic mechanics approach to quantum mechanics of Nelson [N] and also withhomogenization theory for divergencestructure secondorder elliptic PDE.
Action minimizing measures. Hereafter Tn denotes the flat torus in Rn, the unitcube with opposite faces identified. We are given a smooth and periodic potential functionW : Tn R and a vector V Rn. The Lagrangian is
L(v, x) :=12v2 W (x) (v Rn, x Tn),
and the corresponding Hamiltonian is
H(p, x) :=12p2 + W (x) (p Rn, x Tn).
Mathers minimization problem is to find a Radon measure on the velocitypositionconfiguration space Rn Tn to minimize the generalized action
(1.1) A[] :=Rn
Tn
12v2 W d,
subject to the requirements that
(1.2) 0, (Rn Tn) = 1,2

(1.3)Rn
Tn
v D d = 0 for all C1(Tn),
and
(1.4)Rn
Tn
v d = V.
The identity (1.3) is a weak form of flow invariance.
Quantum action minimizing states. We propose as a quantum version of Mathersproblem to find minimizing the action
(1.5) A[] :=Tn
h2
2D2 W 2dx,
subject to the constraints that
(1.6)Tn
2 dx = 1,
(1.7)Tn
(D D) D dx = 0 for all C1(Tn),
and
(1.8)h
2i
Tn
D D dx = V.
Here h denotes a positive constant. We always suppose that has the Bloch wave form
= eiP x
h
for some P Rn and a periodic function : Tn C.
If is smooth, condition (1.7) reads
(1.7) div(D D) = 0.
This is the analogue of the flow invariance. The vector field j := D D representsthe flux.
Remark. While it is presumably possible to introduce some sort of quantization formore general Lagrangians than L(v, x) = 12 v2 W (x), most of the subsequent analysiswould fail: we will from 3 onward rely upon some ColeHopf type transformations thatdepend upon the precise structure of this Lagrangian.
3

2. First and second variations, local minimizers.
In this section we provide some formal calculations concerning the first and second variations of our problem (1.5) (1.8). These heuristic deductions motivate the constructionsand computations in 38.
Let us take the complexvalued state in polar form
(2.1) = aeiu/h,
where the phase u has the structure
(2.2) u = P x + v
for some Tnperiodic function v. Thus has the requisite Bloch wave form. The action isthen
(2.3) A[] =Tn
h2
2Da2 + a
2
2Du2 Wa2 dx,
and the constraints (1.6)(1.8) become
(2.4)Tn
a2 dx = 1,
(2.5) div(a2Du) = 0,
(2.6)Tn
a2Du dx = V.
2.1 First variation. Let {(u(), a())}11 be a smooth oneparameter familysatisfying (2.4)(2.6), with (u(0), a(0)) = (u, a). We suppose also that for each (1, 1),we can write
u() = P () x + v(),
where P () Rn and v() is Tnperiodic. Define
(2.7) j() :=Tn
h2
2Da()2 + a
2()2Du()2 Wa2() dx,
and hereafter write = dd .4

Theorem 2.1. We have j(0) = 0 for all variations if and only if
(2.8) h2
2a = a
( Du22
+ W E)
for some real number E.
We interpret (2.8) as the EulerLagrange equation for our minimization problem, andcall = aeiu/h a critical point if this PDE is satisfied.
Proof. 1. We first compute
j =Tn
h2Da Da + aaDu2 + a2Du Du 2Waa dx.
Next, differentiate (2.5), (2.6):
(2.9) div(2aaDu + a2Du) = 0,
(2.10)Tn
2aaDu + a2Du dx = 0,
and set = 0. Recall also that Du = P + Dv. Multiply (2.9) by v, integrate by parts,then take the inner product of (2.10) with P . Add the resulting expressions to find
Tn
2aaDu2 + a2Du Du dx = 0.
Hence
j(0) =Tn
h2Da Da aaDu2 2Waa dx
= 2Tn
a(h
2
2a
( Du22
+ W)
a
)dx.
Then j(0) = 0 for all such a provided
h2
2a
( Du22
+ W)
a = Ea,
for some real constant E. This is so since the variation a must satisfy the identity
(2.11)Tn
aa dx = 0,
which we obtain upon differentiating (2.4).
Remark. The foregoing deduction depends upon the implict assumption that we canconstruct a wide enough class of variations to permit our concluding (2.8) from the integralindentities involving a. We will return to this point in 8.
2.2 Second variation. We next differentiate j twice with respect to :5

Theorem 2.2. If = aeiu/h is a critical point, then
(2.12) j(0) =Tn
h2Da2 + a2Du2 2(a)2( Du2
2+ W E
)dx.
Proof. 1. We have
(2.13)
j =Tn
h2Da2 + h2Da Da + (a)2Du2
+ aaDu2 + 4aaDu Du + a2Du2
+ a2Du Du 2W (a)2 2Waa dx.Differentiating (2.9), (2.10) again, we find
(2.14) div(2(a)2Du + 2aaDu + 4aaDu + a2Du) = 0
and
(2.15)Tn
2(a)2Du + 2aaDu + 4aaDu + a2Dudx = 0.
Set = 0. Multiply (2.14) by v, integrate, multiply (2.15) by P , and add:
(2.16)Tn
2(a)2Du2 + 2aaDu2 + 4aaDu Du + a2Du Du dx = 0.
2. We employ this equality in (2.13):
j(0) =Tn
2a(h
2
2a a
( Du22
+ W))
+ h2Da2
(a)2Du2 + a2Du2 2W (a)2 dx
=Tn
2a(Ea) + h2Da2 2(a)2( Du2
2+ W
)+ a2Du2 dx
=Tn
h2Da2 + a2Du2 2(a)2( Du2
2+ W E
)dx.
We have used here the identity Tn
aa + (a)2dx = 0,
derived by twice differentiating (2.4).
2.3 Local minimizers. We continue to write = aeiu/h, and now assume as well that
(2.17) a > 0 in Tn.
Observe that this follows from (2.8) and the strong maximum principle, provided a and uare smooth enough.
6

Theorem 2.3. If = aeiu/h is a critical point and (2.17) holds, then
(2.18) j(0) =Tn
a2Du2 + a2D
(a
a
)2
dx > 0,
provided a = 0.
In this case we call = aeiu/h a local minimizer.
Proof. The EulerLagrange equation (2.8) asserts that
h2
2a = a
( Du22
+ W E)
.
Hence
j(0) =Tn
a2Du2 + h2Da2 2(a)2(h
2
2aa
)dx
=Tn
a2Du2 + h2Da2 + h2(a)2 Da2
a2 2h2aDa
Daa
dx
=Tn
a2Du2 + h2a2D
(a
a
)2
dx.
The last term is strictly positive, unless a a for some constant = 0. But this isimpossible, since
Tn
aa dx = 0.
3. Some useful identities.
Motivated by the foregoing calculations, our aim now is constructing an explicit state, which will turn out to be a critical point, and indeed a local minimizer, of A[], subjectto (1.6) (1.8). We start with two linear problems.
3.1 Dual eigenfunctions. Consider the dual eigenvalue problems:
(3.1)
{h22 w + hP Dw Ww = E0w in Tn
w is Tnperiodic
and
(3.2)
{h22 w hP Dw Ww = E0w in Tn
w is Tnperiodic,7

where E0 = E0(P ) R is the principal eigenvalue. Note carefully the minus signs in frontof the potential W . We may assume the real eigenfunctions w, w to be positive in Tn andnormalized so that
(3.3)Tn
wwdx = 1.
Furthermore, we can take w, w and E0 to be smooth in both the variables x and P .We employ a form of the ColeHopf transformation, to define
(3.4){
v := h log wv := h log w.
Then
(3.5){
w = ev/h
w = ev/h,
and a calculation shows that
(3.6){ h2 v + 12 P + Dv2 + W = Hh(P ) in Tn
v is Tnperiodic
and
(3.7){ h
2 v + 12 P + Dv2 + W = Hh(P ) in Tn
v is Tnperiodic,
for
Hh(P ) :=P 22 E0(P ).
Standard PDE estimates applied to (3.6) and (3.7) provide the bounds
(3.8) Dv, Dv C,
for a constant C depending only upon P and the potential W .
3.2 Continuity and eikonal equations. Define
(3.9) := ww
and
(3.10) u := P x + v + v
2.
Note that although w, w, v, v, u and depend on h, we will for notational simplicitymostly not write these functions with a subscript h. The importance of the product (3.9)of the eigenfunctions is noted also in Anantharaman [A].
According to (3.3),
> 0 in Tn,Tn
dx = 1.
8

Theorem 3.1. (i) We have
(3.11) div(Du) = 0 in Tn.
(ii) Furthermore,
(3.12)12Du2 + W Hh(P ) =
h
4(v v) 1
8Dv Dv2 in Tn.
We call (3.11) the continuity (or transport) equation, and regard (3.12) as an eikonalequation with an error term on the right hand side.
Proof. 1. We compute
h div(wDw wDw) = h(ww ww)
=2h
(w
(h2
2w
) w
(h2
2w
))
=2h
(w(E0w Ww + hP Dw)
w(E0w Ww hP Dw))= 2(wP Dw + wP Dw) = 2P D.
But
wDw wDw = w(Dv
hw
) w
(Dv
hw
)= 1
h(Dv + Dv),
and thereforeP D + 1
2div (D(v + v)) = 0.
This is (3.11).
2. Recalling the formula12a b2 + 1
2a + b2 = a2 + b2,
we compute
12
P + 12D(v + v)2
=18(P + Dv) + (P + Dv)2
=14P + Dv2 + 1
4P + Dv2 1
8Dv Dv2.
Hence12Du2 + W Hh(P ) =
12
(12P + Dv2 + W Hh(P )
)
+12
(12P + Dv2 + W Hh(P )
) 1
8D(v v)2
=12
(h
2v
)+
12
(h
2v
) 1
8D(v v)2,
9

owing to (3.6), (3.7).
Remark. We also have the identities
h2 div ((P + Dv)) = 0(3.13)
h2 + div ((P + Dv)) = 0.(3.14)
For a quick derivation, observe first that
h
2 = div
(12D(v v)
).
Add and substract this from (3.11).
3.3 Integral identities involving Du and D2u. To simplify notation, we will hereafter write
(3.15) d := dx.
Theorem 3.2. These formulas hold:
(3.16)Tn
12Du2 + W d = Hh(P ) +
18
Tn
Dv Dv2 d,
(3.17)Tn
D2u2 + 14D2v D2v2 d =
Tn
W d.
Proof. 1. In view of (3.12),Tn
12Du2 + W Hh(P ) d =
h
4
Tn
(v v) dx
18
Tn
Dv Dv2 dx.
But = ww = evv
h , and thereforeTn
12Du2 + W Hh(P ) d =
h
4
Tn
D(v v) D(v v)h
dx
18
Tn
Dv Dv2 dx
=18
Tn
Dv Dv2 d.10

2. Now differentiate the identity (3.12) twice with respect to xk:
Du Duxkxk + Duxk Duxk + Wxkxk =h
4(v v)xkxk
14D(v v) D(v v)xkxk
14D(v v)xk D(v v)xk .
Multiply by and integrate:Tn
Du Duxkxk + Duxk Duxk + Wxkxk d
=h
4
Tn
(v v)xkxk d 14
Tn
D(v v) D(v v)xkxk d
14
Tn
D(v v)xk D(v v)xk d.
According to Theorem 3.1, the first term on the left vanishes. We integrate by parts inthe first term on the right, and thereby derive an expression that cancels the second termon the right. Next sum on k, to derive (3.17).
4. First and second derivatives of Hh.
As explained in [EG], the behavior of various expressions as functions of P is important:
Theorem 4.1. We have
(4.1) DHh(P ) =Tn
Du d,
(4.2) D2Hh(P ) =Tn
D2xP uD2xP u d +14
Tn
D2xP (v v)D2xP (v v) d.
In particular, Hh is a convex function of P .
Our notation means that the (l, m)th component of the first term on the right hand sideof (4.2) is
Tn
uxiPluxiPm d and of the second term is14
Tn
(v v)xiPl(v v)xiPm d.
Proof. 1. We differentiate (3.1) and (3.2) with respect to Pk:
(4.3) h2
2wPk + hP DwPk WwPk E0wPk = E0Pkw hwxk ,
(4.4) h2
2wPk hP Dw
PkWwPk E
0wPk = E0Pk
w + hwxk .11

Multiply (4.3) by w and integrate over Tn. Since w solves (3.2), we deduce
DE0(P ) = hTn
Dwwdx = Tn
Dv wwdx = Tn
Dv d.
Similarly, we multiply (4.4) by w and integrate:
DE0(P ) = hTn
Dww dx = Tn
Dv d.
Then
DHh(P ) = P DE0(P ) = P +12
Tn
Dv + Dv d =Tn
Du d.
2. Next, differentiate the identity (3.12) with respect to Pk and Pl:
Hh,PkPl = Du DuPkPl + DuPk DuPl h
4(v v)PkPl
+14D(v v) D(v v)PkPl +
14D(v v)Pk D(v v)Pl .
Multiply by and integrate, to discover
Hh,PkPl =Tn
Du DuPkPl + DuPk DuPl d h
4
Tn
(v v)PkPl d
+14
Tn
D(v v) D(v v)PkPl d +14
Tn
D(v v)Pk D(v v)Pl d.
In view of Theorem 3.1 and the periodicity of uPkPl , the first term on the right vanishes.Since = ww = e
vvh , we can integrate by parts in the third term on the right, obtaining
an expression that cancels the fourth term. Formula (4.2) results.
As an application of Theorem 4.1, we modify some ideas from [EG] and [E2] to discussthe effects of a nonresonance condition on the asymptotics as h 0.
We will explain in the next section that as h 0 the functions Hh converge uniformlyon compact sets to the convex function H, the effective Hamiltonian in the sense of LionsPapanicolaouVaradhan [LPV]. Let us suppose that H is differentiable at P and thatV = DH(P ) satisfies
(4.5) V m = 0 for each m Zn, m = 0.
12

Theorem 4.2. Suppose also that D2Hh(P ) is bounded as h 0. Then
(4.6) limh0
Tn
(DP u) d =Tn
(X) dX
for each continuous, Tnperiodic function .
We discuss in [EG] that this statement is consistent with the classical assertion that theHamiltonian dynamics in the X, P variables, where X := DP u, correspond to the trivialmotion X = V, P = 0.
Proof. Fix any m Zn, and observe that the function
e2imDP u = e2imxeimDP (v+v)
is Tnperiodic. Consequently
(4.7)
0 =Tn
Du D(e2imDP u
)d = 2i
Tn
e2imDP umkuxj uxjPk d
= 2iTn
e2imDP umkHh,Pk d + 2iTn
e2imDP umk(uxj uxjPk Hh,Pk) d
=: 2i(A + B).
We claim now that
(4.8) B = O(h) as h 0.
To confirm this, notice first that our differentiating (3.12) gives the identity
uxj uxjPk Hh,Pk =h
4(v v)Pk
14(v v)xj (v v)xjPk ;
and therefore(uxj uxjPk Hh,Pk) =
h
4((v v)xjPk
)xj
.
HenceB =
Tn
e2imDP umk(uxj uxjPk Hh,Pk) dx
=Tn
e2imDP umkh
4((v v)xjPk
)xj
dx
= hi2
Tn
e2imDP umluxjPlmk(v v)xjPk d.
SoB Ch
Tn
D2xP u2 + D2xP (v v)2 d = O(h)13

according to (4.2), since D2Hh(P ) is bounded. This proves (4.8).But then (4.7) implies
m DHh(P )Tn
e2imDP u d = B O(h).
We will see in 5 that Hh H, locally uniformly. Since therefore DHh(P ) DH(P ) = Vand since m V = 0, we deduce that
limh0
Tn
e2imDP u d = 0.
This limit holds for all m = 0, and hence
limh0
Tn
(DP u) d =Tn
(X) dX
for each Tnperiodic function .
5. An identity involving exact solutions of the eikonal equation.
Assume next that v is a Lipschitz continuous almost everywhere solution of the cellproblem
(5.1){ 1
2 P + Dv2 + W = H(P ) in Tnv is Tnperiodic.
The term on the right hand side of (5.1) is the effective Hamiltonian in the sense of LionsPapanicolaouVaradhan [LPV], a central assertion of which is that for a given vectorP problem (5.1) is solvable in the sense of viscosity solutions. (Our H corresponds toMathers function , and is equivalent also to Manes constant.)
Write
(5.2) u := P x + v;
so that
(5.3)12Du2 + W = H(P ) almost everywhere in Tn.
14

Theorem 5.1. This formula holds:
(5.4)12
Tn
12D(v + v)Dv2 d + 1
8
Tn
Dv Dv2 d = H(P ) Hh(P ).
Proof. We employ the identity
12a b2 + a (b a) = 1
2b2 1
2a2
witha = P +
12D(v + v) = Du, b = P + Dv = Du,
to discover
12
Tn
12D(v + v)Dv2 d +
Tn
Du D(v 12(v + v)) d
=Tn
12Du2 + W d
Tn
12Du2 + W d.
Owing to Theorem 3.1 and the periodicity of v, v and v, the second term on the leftvanishes. Consequently (5.3) and (3.16) imply
12
Tn
12D(v + v)Dv2 d = H(P ) Hh(P )
18
Tn
Dv Dv2 d.
As an application, we have the estimate:
Theorem 5.2. (i) For each P Rn,
(5.5) Hh(P ) H(P ) Hh(P ) + O(h) as h 0.
(ii) Hence
(5.6)Tn
12D(v + v)Dv2 d +
Tn
Dv Dv2 d = O(h).
Proof. 1. According to (5.4), Hh(P ) H(P ).In addition, we have the minimax formula
(5.7) H(P ) = infvC1(Tn)
maxxTn
{12P + Dv2 + W (x)
}.
15

(See for instance the appendix of [E2] for a quick proof due to A. Fathi.) Furthermore,standard PDE estimates deduce from (3.6), (3.7) the onesided second derivative bounds
(5.8) v C and v C,
for some constant C and any unit vector . Then formula (3.12) implies
12P + 1
2D(v + v)2 + W Hh(P ) + Ch.
Consequently, we can deduce from (5.7) that
H(P ) maxxTn
{12P + 1
2D(v + v)2 + W (x)
} Hh(P ) + Ch.
2. Statement (ii) follows from (5.4), (5.5)
6. Quantum Lagrangian calculations.
In this section we draw some connections between the minimization problems, bothclassical and quantum, discussed in 12, and the explicitly constructed state studiedin 35.
6.1 Quantum action. As before, we have a = 1/2 = evv
2h and we continue to write = ae
iuh . Recall as well that the action of is
A[] :=Tn
h2
2D2 W 2 dx.
We next demonstrate that satisfies the EulerLagrange equation (2.8).
Theorem 6.1. We have
(6.1)12Du2 + W Hh(P ) =
h2
2aa
in Tn.
In particular, is a critical point of the action A[], subject to (1.6) (1.8).According to Theorem 2.3, is a local minimizer as well. I conjecture that is in fact
a global minimizer, but am unable to prove this.
Proof. Since a = evv
2h , we compute
h2
2aa
= h2
2e
vv2h (e
vv2h )
= h2
2e
vv2h (
12h
(v v) + 14h2D(v v)2)e v
v2h
=h
4(v v) 1
8D(v v)2
=12Du2 + W Hh(P ),
16

the last equality being (3.12).
We now compute the action of . To do so, we first introduce Lh, the Legendre transformof Hh, and also write
(6.2) Vh := DHh(P ) =Tn
Du d.
Theorem 6.2. The quantum action of is
(6.3) A[] = Lh(Vh).
Proof. Let us employ (3.11) and (3.13), to deduce
A[] =Tn
h2
2D2 W 2dx
=Tn
h2
2Da2 + a
2
2Du2 Wa2dx
=Tn
h2
2Da2 dx +
Tn
Du2a2 dxTn
(12Du2 + W
)a2 dx
=18
Tn
Dv Dv2 d +Tn
(P +12D(v + v)) Du d
Tn
12Du2 + W d
= P Vh Hh(P ) = Lh(Vh).
6.2 Convergence as h0. We now determine the behavior of and u, defined by(3.9), (3.10), as h 0. For the remainder of this section we for clarity add subscripts hto display the dependence on this parameter. Thus
h = whwh, uh = P x +vh + vh
2, etc.
Define a measure h on velocityposition configuration space by requiring
(6.4)Rn
Tn
(v, x) dh :=Tn
(Duh(x), x) dh
for each continuous : Rn Tn R. In view of estimate (3.8) the measures {h}h>0have uniformly bounded support, and we can consequently obtain a sequence hj 0 anda probabilitly measure such that
hj weakly as measures.
We may also suppose that
(6.5) Vhj V in Rn.
17

Theorem 6.3. The measure solves Mathers minimization problem (1.1) (1.4).
Proof. 1. First we check that satisfies the constraints (1.2) (1.4), the first of which isclear since is a probability measure. If furthermore C1(Tn), then
Rn
Tn
v D d = limj
Tn
Duhj D dhj = 0,
since div(hDuh) = 0. This is (1.3); and (1.4) similarly holds sinceRn
Tn
v d = limj
Tn
Duhj dhj = limj
Vhj = V.
2. Recall from (5.5) that Hh H, uniformly on compact sets. Therefore Theorem 6.2and (6.5) imply
(6.6)
L(V ) = limj
Lhj (Vhj )
= limj
Tn
h2j2Dahj 2 +
a2hj2Duhj 2 Wa2hj dx
= limj
18
Tn
Dvhj Dvhj 2 dhj + lim
j
Tn
12Duhj 2 W dhj
= limj
Rn
Tn
12v2 W dhj
=Rn
Tn
12v2 W d = A[].
Here we recalled (5.6) to ensure that the first term on the third line goes to 0.
3. Suppose now is any other measure satisfying (1.2) (1.4). Let u = u, where is a standard mollifier and as before u solves the eikonal equation (5.3) for any P L(V ).Then
12Du2 + W H(P ) + C.
for some constant C, everywhere on Tn. Therefore
A[] =Rn
Tn
12v2 W d
Rn
Tn
12v2 + 1
2Du2 d H(P ) C.
Rn
Tn
v Du d H(P ) C.
=Rn
Tn
v Dv d + P V H(P ) C.
= L(V ) C..18

This holds for each . > 0, and so L(V ) is less than or equal to the action of any measure satisfying (1.2) (1.4). But then (6.6) guaranties that the measure is a minimizer.
Anantharaman [A] provides interesting and additional information about the limit minimizing measure.
Limits of the eikonal equations. Upon passing if necessary to a further subsequence,we deduce from (3.6), (3.7) that
vhj v, vhj v uniformly on Tn,
where v, v are viscosity solutions of the respective PDE
(6.7)12P + Dv2 + W = H(P ) in Tn
and
(6.8) 12P + Dv2 W = H(P ) in Tn.
In particular the Lipschitz continuous functions v, v solve (6.7), (6.8) almost everywhere.Therefore
uhj u := P x +v + v
2uniformly on Tn,
where
(6.9)12Du2 + W H(P ) almost everywhere.
Finally, write
:= projx .
for the projection onto Tn. Then
hj weakly as measures.
Applying the regularity theory from [EG], we deduce that Dv, Dv, and therefore Du,exist for each point in spt, and
(6.10)12Du2 + W = H(P ) on spt.
19

7. Quantum Hamiltonian calculations.
7.1 Quasimodes.The observations in the last section show that our construction in 3 is a sort of semi
classical quantization of weak KAM theory.We next show that built above is an approximate solution of the stationary Schrodinger
equation
h2
2 + W = E in Tn.
Notice the plus sign in front of the potential W . As usual, a = 1/2 = evv
2h and = aeiuh .
Then
(7.1)h
2
2 + W E =
(12Du2 + W E
) ih
2div(a2Du)
a2 h
2
2aa
=: A + B + C.
In view of Theorem 3.1, B 0. Now take
E = Hh(P ).
According to Theorem 6.1,A C;
that is, the formal O(1)term identically equals the formal O(h2)term in the expansion(7.1). Therefore
(7.2) h2
2 + W E = 2(1
2Du2 + W Hh(P ))
for E = Hh(P ). It is sometimes useful to rewrite this as
(7.3) h2
2 = (Du2 + W Hh(P )).
Theorem 7.1. If E = Hh(P ),
h2
2 + W E = O(h),
the right hand side estimated in L2(Tn).
Proof. Define the remainder term
(7.4) R := 2(12Du2 + W Hh(P )).
20

Then
14
Tn
R2 dx =Tn
(h
4(v v) 1
8D(v v)2
)2d
=Tn
h2
16((v v))2 h
16(v v)D(v v)2 + 1
64D(v v)4 d.
Observe now that
h16
Tn
(v v)D(v v)2 d = h16
Tn
(v v)D(v v)2e vvh dx
=116
Tn
D(v v)2D(v v) D(v v) d
+h
8
Tn
D(v v) D2(v v)D(v v) d
= 116
Tn
D(v v)4 d
+h
8
Tn
D(v v) D2(v v)D(v v) d.
Since 116 >164 , we derive for some constant C the estimate:
(7.5)Tn
R2 dx +Tn
D(v v)4 d Ch2Tn
D2(v v)2 d.
We deduce finally from (7.5) and (3.17) that R is of order at most O(h) in L2(Tn).
Remark. The O(h)error term is not especially good, and indeed M. Zworski hasoutlined for me some other constructions building quasimodes with similar error estimatesin quite general circumstances. Estimate (7.5) does show that if E = Hh(P ) and if
(7.6)Tn
D2(v v)2 d = o(1),
we would then have the better error bound
h2
2 + W E = o(h) as h 0
in L2(Tn). We may hope that assertion (7.6) is true in some generality, although it canfail, as the following shows:
Example. Assume that P = 0 and that the potential W attains its maximum at aunique point x0 Tn, where W (x0) < 0. In this situation we can readily check thatH(0) = W (x0).
21

Since P = 0, we can take w w; whence v v and so u 0. Then according to(3.16) and (5.15),
limh0
Tn
W d = W (x0) = maxTn
W.
So the weak limit of the measures as h 0 is the unit mass at x0. Consequently, theidentity (3.17) implies
limh0
Tn
D2(v v)2 d = 4W (x0) > 0.
This example is from the forthcoming paper of Y. Yu [Y], who provides a very completeanalysis of our problem in one dimension. In particular, if n = 1 and H(P ) > min H, theerror term is indeed o(h) in L2 as h 0.
7.2 Comparison with stochastic mechanics There are formal connections with theGuerraMorato and Nelson variational principle in stochastic quantum mechanics, as setforth in Nelson [N], GuerraMorato [GM], Yasue [Ya], Carlen [C], etc. (I thank A. Majdafor some of these references.)
I attempt here to explain the link by recasting their form of the action into our settingand notation, disregarding the probabilistic interpretations. In effect, then, the action ofGuerraMorato becomes
(7.10) A[] :=Tn
(12Dv Dv W
)a2 dx
for = aeiu/h, a = evv
2h , u = 12 (v+v), P = 0. (The Lagrangian density
(12Dv
Dv W)a2
here should be compared with formula (93) in GuerraMorato [GM]. See also (14.31) inNelson [N].) We rewrite this, observing that
h2
2Da2 + a
2
2Du2 = a
2
2Dv Dv.
Consequently
(7.11) A[] =Tn
h2
2Da2 + a
2
2Du2 Wa2 dx.
This action differs from ours due to the sign change in the first term.
Theorem 7.2. Let = aeiu/h be a smooth critical point of the action A[], subject to theconstants (1.6)(1.8). Then for some real constant E:
(7.12) h2
2 + W = E in Tn,
22

and so is an exact solution of this stationary Schrodinger equation.
Proof. We deduce, as in the proof of Theorem 2.1, that
h2
2a + a
( Du22
+ W E)
= 0,
with a sign change as compared with (2.8). Thus if we write out (7.1), we have
h2
2 + W E = A + B + C,
and B 0, A + C 0.
8. Connections with linear elliptic homogenizationThis section works out some relationships between Hh and homogenization theory
for divergencestructure, second order elliptic PDE: see for instance BensoussanLionsPapanicolaou [BLP]. Our conclusions are very similar to those of Capdeboscq [Cp], andthe calculations in Pedersen [P] are related as well.
Let A = ((aij)) be symmetric, positive definite, and Tnperiodic. Suppose U is abounded, open subset of Rn, with a smooth boundary. We consider this boundary valueproblem for an elliptic PDE with rapidly varying coefficients:
{
(aij
(x
)uxi
)xj
= f in U
u = 0 on U.
Then u u weakly in H10 (U), u solving the limit problem{ aijuxixj = f in Uu = 0 on U.
The effective diffusion coefficient matrix A = ((aij)) is determined as follows [BLP]. Forj = 1, . . . , n, let j solve the corrector problem
(8.1){ (akljxk)xl = (ajl)xl in Tn
j is Tnperiodic.
Let us then for i, j = 1, . . . , n define
(8.2) aij :=Tn
aij aklixkjxl
dx.
23

Theorem 8.1. Define Vh = DHh(P ). Then
(8.3) AP = Vh,
where A = ((aij)) is the effective diffusion coefficient matrix corresponding to
(8.4) A := a2I = ((a2ij)).
Notice in (8.4) that a2 = = ww depends on both h and P .
Proof. For the special case of the diagonal matrix A given by (8.4), the corrector PDE(8.1) reads
(8.5){ (a2jxk)xk = (a2)xj in Tn
j is Tnperiodic.
Now u = x P + 12 (v + v) solves
div(Du) = 0,
and therefore
div(
a212D(v + v)
)= div(a2P ) = D(a2) P.
Hence
(8.6)v + v
2= Pii.
Consequently, for j = 1, . . . , n
Vh,j =Tn
uxj d =Tn
(i + xi)xj Pi d
= Pj Tn
iPixj dx = Pj +Tn
iPi(a2jxk)xk dx
= Pj Tn
Pia2ixk
jxk
dx = (AP )j .
Remark: constructing a family of variations. We return finally to a point left openin 2. Recall that we introduced a one parameter family of variations {(u(), a())}11satisfying the constraints (2.4)(2.6), with (u(0), a(0)) = (u, a). We assumed also that
(8.7) u() = P () x + v(),24

for P () Rn and v() is Tnperiodic.To finish up the proof of Theorem 2.1 we needed to know that a := dad was arbitrary,
subject only to the integral identity (2.11). We show next we can indeed do so, provided
(8.8) a > 0 in Tn.
To confirm this, take a smooth function a() of satisfying (2.4) and a(0) = a. For agiven P Rn, we invoke the Fredholm alternative to solve
div(a2()Dv) = div(a2()P )
for a periodic function v = v(). Then u() = P x + v() solves div(a()2Du()) = 0,and the issue is whether for small we can select P = P () so that
(8.9)Tn
a2()Du() dx = V.
We next introduce the function = {1, . . . ,n} defined by
(P ) :=Tn
a2Du dx = P +Tn
a2Dv dx.
Now since div(a2Dv) = div(a2P ), we have
(a2vxkPl)xk = (a2)xl ;
and sovPl =
l
in the notation above. Therefore, as in the proof of Theorem 8.1, we can calculate
kPl = kl +Tn
a2vxkPl dx = kl Tn
(a2)xkvPl dx
= kl +Tn
(a2vxiPk)xivPl dx = kl Tn
a2vxiPkvxiPl dx
= akl.
In other words, D(P ) = A and the latter matrix is nonsingular. Hence the ImplicitFunction Theorem ensures for small that we can find P = P () satisfying (8.9).
25

References
[A] N. Anantharaman, Gibbs measures on path space and viscous approximation to actionminimizing
measures, Transactions of the AMS (2003).
[BLP] A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures,
NorthHolland, 1978.
[Cp] Y. Capdeboscq, Homogenization of a neutronic critical diffusion problem with drift, Proc. Royal
Soc Edinburgh (to appear).
[C] E. Carlen, Conservative diffusions, Comm Math Physics 94 (1984), 293315.
[E1] L. C. Evans, Effective Hamiltonians and quantum states, Seminaire Equations aux Derivees Par
tielles, Ecole Polytechnique (20002001).
[E2] L. C. Evans, Some new PDE methods for weak KAM theory, to appear in Calculus of Variations.
[EG] L. C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian Dynamics I,
Archive for Rational Mechanics and Analysis 157 (2001), 133.
[F1] A. Fathi, Theoreme KAM faible et theorie de Mather sur les systemes lagrangiens, C. R. Acad.
Sci. Paris Sr. I Math. 324 (1997), 10431046.
[F2] A. Fathi, Solutions KAM faibles conjuguees et barrieres de Peierls, C. R. Acad. Sci. Paris Sr. I
Math. 325 (1997), 649652.
[F3] A. Fathi, Weak KAM theory in Lagrangian Dynamics, Preliminary Version, Lecture notes (2001).
[G] D. Gomes, A stochastic analogue of AubryMather theory, Nonlinearity 15 (2002), 581603.
[GM] F. Guerra and L. Morato, Quantization of dynamical systems and stochastic control theory, Phys
ical Review D 27 (1983), 17741786.
[HK] D. Holcman and I. Kupka, Singular perturbations and first order PDE on manifolds, preprint
(2002).
[LPV] P.L. Lions, G. Papanicolaou, and S. R. S. Varadhan, Homogenization of HamiltonJacobi equa
tions, unpublished, circa 1988.
[M1] J. Mather, Minimal measures, Comment. Math Helvetici 64 (1989), 375394.
[M2] J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math.
Zeitschrift 207 (1991), 169207.
[MF] J. Mather and G. Forni, Action minimizing orbits in Hamiltonian systems, Transition to Chaos
in Classical and Quantum Mechanics, Lecture Notes in Math 1589 (S. Graffi, ed.), Sringer, 1994.
[N] E. Nelson, Quantum Fluctuations, Princeton University Press, 1985.
[P] F. B. Pedersen, Simple derivation of the effective mass equation using a multiplescale technique,
European Journal of Physics 18 (1997), 4345.
[Ya] K. Yasue, Stochastic calculus of variations, Journal of Functional Analysis 41 (1981), 327340.
[Y] Y. Yu, Error estimates for a quantization of a Mather set (to appear).
Department of Mathematics, University of California, Berkeley, CA 94720
26