Adiabatic limits and problems on distribution of
integer points
Yuri A. Kordyukov
Institute of Mathematics
Russian Academy of Sciences
Ufa, Russia
August, 2010
Yuri A. Kordyukov (Ufa, Russia)
Adiabatic limits and integer points
Moscow, Delone120, 2010
1 / 24
Adiabatic limits
The setting
Adiabatic limits
(M, gM ) a smooth compact Riemannian manifold.
F a smooth foliation on M.
F = T F the tangent bundle of F, H = F ⊥ :
M
TM = F
H.
The corresponding decomposition of the metric:
gM = gF + gH .
gε the one-parameter family of Riemannian metrics on M
gε = gF + ε−2 gH ,
Yuri A. Kordyukov (Ufa, Russia)
Adiabatic limits and integer points
ε > 0.
Moscow, Delone120, 2010
2 / 24
Adiabatic limits
The setting
Adiabatic limits
Definition
The Laplace-Beltrami operator defined by gε , ε > 0:
∆ε = dg∗ε d,
where
d : C ∞ (M) → Ω1 (M) is the de Rham differential;
dg∗ε : Ω1 (M) → C ∞ (M) the adjoint of d with respect to the inner
products defined by gε .
Problem
The limit ε → 0 — the adiabatic limit.
Yuri A. Kordyukov (Ufa, Russia)
Adiabatic limits and integer points
Moscow, Delone120, 2010
3 / 24
Adiabatic limits
The main problem
The problem
For ε > 0, ∆ε a second order self-adjoint elliptic operator on the
compact manifold M =⇒ it has a complete orthonormal system of
eigenfunctions
∆ε ϕj (ε) = λj (ε)ϕj (ε),
ϕj (ε) ∈ C ∞ (M),
where λ0 (ε) = 0 < λ1 (ε) ≤ λ2 (ε) ≤ . . ., λj (ε) → ∞ as j → ∞.
Problem
The asymptotic behavior of the eigenvalue distribution function
Nε (λ) = #{j : λj (ε) < λ},
ε → 0.
or, more generally, of
tr f (∆ε ) =
X
f (λj (ε)),
ε → 0.
j
Yuri A. Kordyukov (Ufa, Russia)
Adiabatic limits and integer points
Moscow, Delone120, 2010
4 / 24
Linear foliation on the two-torus
Preliminaries
Linear foliations on the two-torus
T2 = R2 /Z2 the two-dimensional torus with the coordinates
(x, y ) ∈ R2 , considered modulo integer translations;
g the Euclidean metric on T2 :
g = dx 2 + dy 2 .
F the foliation on T2 determined by the parallel lines
e(x ,y ) = {(x0 + t, y0 + tα) ∈ R2 : t ∈ R},
L
0 0
(x0 , y0 ) ∈ R2 ,
with the slope α ∈ R.
Yuri A. Kordyukov (Ufa, Russia)
Adiabatic limits and integer points
Moscow, Delone120, 2010
5 / 24
Linear foliation on the two-torus
Preliminaries
The adiabatic limit
The Riemannian metric gε is given by
gε =
1
ε−2
2
(dx
+
αdy
)
+
(−αdx + dy )2 .
1 + α2
1 + α2
The corresponding Laplace operator:
1
∆ε = −
1 + α2
Yuri A. Kordyukov (Ufa, Russia)
∂
∂
+α
∂x
∂y
2
ε2
−
1 + α2
Adiabatic limits and integer points
∂
∂
−α
+
∂x
∂y
2
Moscow, Delone120, 2010
.
6 / 24
Linear foliation on the two-torus
Preliminaries
The eigenvalue distribution function
Eigenvalues and eigenfunctions
The operator ∆ε has a complete orthogonal system of eigenfunctions
ukl (x, y ) = e2πi(kx+ly ) ,
(x, y ) ∈ T2 ,
with the corresponding eigenvalues
1
ε2
2
2
2
(k + αl) +
(−αk + l) ,
λkl (ε) = (2π)
1 + α2
1 + α2
(k , l) ∈ Z2 .
The eigenvalue distribution function of ∆ε
Nε (λ) = the number of integer points in the ellipse
1
ε2
2
2
2
2
{(ξ, η) ∈ R : (2π)
(ξ + αη) +
(−αξ + η) < λ}.
1 + α2
1 + α2
Yuri A. Kordyukov (Ufa, Russia)
Adiabatic limits and integer points
Moscow, Delone120, 2010
7 / 24
Linear foliation on the two-torus
The main result
The asymptotic formula
Theorem (A. Yakovlev, 2007)
1. For α 6∈ Q and λ ∈ R
Nε (λ) =
2. For α =
p
q
Nε (λ) = ε−1
1 −1
ε λ + o(ε−1 ),
4π
ε → 0.
∈ Q, where p ∈ Z and q ∈ Z are coprime, and λ ∈ R
X
2
k : 24π 2 k 2 <λ
p +q
Yuri A. Kordyukov (Ufa, Russia)
1
4π 2
p
(λ− 2
k 2 )1/2 +o(ε−1 ),
2
2
2
p
+
q
π p +q
Adiabatic limits and integer points
ε → 0.
Moscow, Delone120, 2010
8 / 24
Linear foliation on the two-torus
Noncommutative Weyl formula
The semiclassical Weyl formula
M a compact manifold;
V ∈ C ∞ (M, R) a real-valued smooth function;
the Schrödinger operator
Hh = h2 ∆ + V (x),
x ∈ M.
the semiclassical principal symbol of Hh :
p(x, ξ) = |ξ|2 + V (x),
(x, ξ) ∈ T ∗ M.
The semiclassical Weyl formula:
Z
1
tr f (Hh ) =
f (p(x, ξ)) dx dξ + o(h−n ),
(2πh)n T ∗ M
Yuri A. Kordyukov (Ufa, Russia)
Adiabatic limits and integer points
h →0+.
Moscow, Delone120, 2010
9 / 24
Linear foliation on the two-torus
Noncommutative Weyl formula
The principal symbol
The operator
1
∆ε = −
1 + α2
∂
∂
+α
∂x
∂y
2
ε2
−
1 + α2
∂
∂
−α
+
∂x
∂y
2
.
The conormal bundle
N ∗ F = {(x, y , p2 ) ∈ T2 × R}.
The lifted foliation FN on N ∗ F is defined by the orbits of the
induced flow on N ∗ F
Tτ (x, y , p2 ) = (x + τ, y + ατ, p2 ),
(x, y , p2 ) ∈ T2 × R.
The principal symbol of ∆ε is a tangentially elliptic operator in
C ∞ (N ∗ F) given by
1
∂
∂ 2
σ(∆ε ) = −
+α
+ p22 .
∂y
1 + α2 ∂x
Yuri A. Kordyukov (Ufa, Russia)
Adiabatic limits and integer points
Moscow, Delone120, 2010
10 / 24
Linear foliation on the two-torus
Noncommutative Weyl formula
Operator algebras
Now we assume that α 6∈ Q.
The restriction of the operator σ(∆ε ) to
L̃(x,y ,p2 ) = {(x + τ, y + ατ, p2 ) ∈ N ∗ F : τ ∈ R} ∼
=R
is the √
second order elliptic differential operator in the space
L2 (R, 1 + α2 dτ ):
σ(∆ε )(x,y ,p2 ) = −
1
∂2
+ p22 .
1 + α2 ∂τ 2
The family σ(∆ε ) = {σ(∆ε )(x,y ,p2 ) : (x, y , p2 ) ∈ N ∗ F} is affiliated
with a certain C ∗ -algebra C ∗ (N ∗ F, FN ), called the foliation
C ∗ -algebra of (N ∗ F, FN ).
The family e−tσ(∆ε ) = {e−tσ(∆ε )(x,y ,p2 ) : (x, y , p2 ) ∈ N ∗ F} belongs
to C ∗ (N ∗ F, FN ),
Yuri A. Kordyukov (Ufa, Russia)
Adiabatic limits and integer points
Moscow, Delone120, 2010
11 / 24
Linear foliation on the two-torus
Noncommutative Weyl formula
Integration over N ∗ F/FN
The foliation (N ∗ F, FN ) has a natural transverse symplectic
structure
=⇒ it has a natural holonomy invariant transverse measure (a
transverse Liouville measure).
By noncommutative integration theory, there exists the
corresponding trace trFN on C ∗ (N ∗ F, FN ).
One can show that
trFN e−tσ(∆ε ) < ∞.
Yuri A. Kordyukov (Ufa, Russia)
Adiabatic limits and integer points
Moscow, Delone120, 2010
12 / 24
Linear foliation on the two-torus
Noncommutative Weyl formula
Integration (continued)
√
The kernel of e−tσ(∆ε )(x,y ,p2 ) in L2 (R, 1 + α2 dτ )
(τ1 − τ2 )2
−1/2 −p22 t
Kt (τ1 , τ2 ) = (4πt)
e
exp −
.
4t(1 + α2 )
Putting τ1 = τ2 = 0, we get a well-defined function kt on N ∗ F:
2
kt (x, y , p2 ) = (4πt)−1/2 e−p2 t ,
(x, y , p2 ) ∈ T2 × R.
Finally, we obtain
trFN e
−tσ(∆ε )
Yuri A. Kordyukov (Ufa, Russia)
Z
=
T2 ×R
kt (x, y , p2 ) dx dy dp2 =
Adiabatic limits and integer points
1
.
2t
Moscow, Delone120, 2010
13 / 24
Linear foliation on the two-torus
Noncommutative Weyl formula
Noncommutative Weyl formula
Fact
tr e
−t∆ε
Z
+∞
=
0
e−tλ dNε (λ) =
1 −1
ε + o(ε−1 ),
4πt
ε → 0.
Theorem
tr e−t∆ε =
Yuri A. Kordyukov (Ufa, Russia)
1
trF e−tσ(∆ε ) + o(ε−1 ),
2πε N
Adiabatic limits and integer points
ε → 0.
Moscow, Delone120, 2010
14 / 24
Linear foliation on the two-torus
Noncommutative Weyl formula
Some remarks
Remark
For α ∈ Q: the orbit through (x, y , p2 ):
L̃(x,y ,p2 ) = {(x + τ, y + ατ, p2 ) ∈ N ∗ F : τ ∈ R} ∼
= S1.
Therefore, the restriction of σ(∆ε ) to L̃(x,y ,p2 ) is the second order elliptic
√
differential operator σ(∆ε )(x,y ,p2 ) in the space L2 (S 1 , 1 + α2 dτ ). So
we get a sum over Z.
Remark
Similar formula holds for any Riemannian foliation (Yu. K., 1999).
Yuri A. Kordyukov (Ufa, Russia)
Adiabatic limits and integer points
Moscow, Delone120, 2010
15 / 24
Integer points in domains
Preliminaries
The setting
F a p-dimensional linear subspace of Rn ;
H = F ⊥ the q-dimensional orthogonal complement of F with
respect to the standard inner product (·, ·) in Rn , p + q = n;
For any ε > 0, consider the linear transformation Tε : Rn → Rn :
(
x,
if x ∈ F ,
Tε (x) =
−1
ε x, if x ∈ H.
For any bounded domain S in Rn with smooth boundary, we put
nε (S) = #(Tε (S) ∩ Zn ),
ε > 0.
Problem
The asymptotic behavior of nε (S) as ε → 0.
Yuri A. Kordyukov (Ufa, Russia)
Adiabatic limits and integer points
Moscow, Delone120, 2010
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Integer points in domains
Preliminaries
Some auxiliary notions
Γ = Zn ∩ F a free abelian group (r = rank Γ ≤ p the rank of Γ)
For r ≥ 1, denote by (`1 , `2 , . . . , `r ) a base in Γ.
V the r -dimensional subspace of Rn spanned by (`1 , `2 , ...`r )
(Γ is a lattice in V !).
Γ∗ denote the lattice in V , dual to the lattice Γ:
Γ∗ = {γ ∗ ∈ V : (γ ∗ , Γ) ⊂ Z}.
For any x ∈ V , we denote by Px the (n − r )-dimensional affine
subspace of Rn , passing through x orthogonal to V .
Fact
Γ∗ coincides with the orthogonal projection of Zn to V :
[
Zn ⊂
(Pγ ∗ ∩ S).
γ ∗ ∈Γ∗
Yuri A. Kordyukov (Ufa, Russia)
Adiabatic limits and integer points
Moscow, Delone120, 2010
17 / 24
Integer points in domains
Preliminaries
Some auxiliary notions
Q ⊂ V the parallelepiped spanned by the base (`1 , `2 , ...`r ) in Γ.
|Q| the r -dimensional Euclidean volume of Q:
|Q| = volr (`1 , `2 , ...`r ) = vol(V /Γ).
Remark: for r = 0, the groups Γ and Γ∗ are trivial, it is natural to
put |Q| = 1.
Yuri A. Kordyukov (Ufa, Russia)
Adiabatic limits and integer points
Moscow, Delone120, 2010
18 / 24
Integer points in domains
The results
The first result
Theorem (Yu.K., A. Yakovlev, 2010)
For any bounded open set S in Rn with smooth boundary, the formula
holds:
nε (S) =
1
ε−q X
−q
voln−r (Pγ ∗ ∩ S) + O(ε p−r +1 ),
|Q| ∗ ∗
ε → 0.
(1)
γ ∈Γ
Remark
In the case when F is trivial, we have p = r = 0, q = n. The problem is
reduced to the classical problem on the asymptotics of the number of
integer points in a family of homothetic domains in Rn . Our formula is
reduced to the classical formula, going back to Gauss:
#(ε−1 S ∩ Zn ) = ε−n voln (S) + O(ε1−n ),
Yuri A. Kordyukov (Ufa, Russia)
Adiabatic limits and integer points
ε → 0.
Moscow, Delone120, 2010
19 / 24
Integer points in domains
The results
The second result
Theorem (Yu.K., A. Yakovlev, 2010)
For any bounded open set S in Rn with smooth boundary such that, for
any x ∈ F , the intersection S ∩ {x + H} is strictly convex, the formula
holds:
nε (S) =
ε−q X
voln−r (Pγ ∗ ∩ S) + O(εk −q ),
|Q| ∗ ∗
ε → 0,
(2)
γ ∈Γ
where
(
k=
Yuri A. Kordyukov (Ufa, Russia)
q+1
2(p−r +1)
2q
q+1+2(p−r )
if
if
q−1
2
q−1
2
≤p−r
> p − r.
Adiabatic limits and integer points
Moscow, Delone120, 2010
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Integer points in domains
The results
Remark
In the case when F is trivial, we get k = 2n/(n + 1), and the formula is
reduced to the following formula:
2
#(ε−1 S ∩ Zn ) = ε−n voln (S) + O(ε−n+2− n+1 ),
ε → 0.
This formula was proved by Randol in 1966.
Yuri A. Kordyukov (Ufa, Russia)
Adiabatic limits and integer points
Moscow, Delone120, 2010
21 / 24
Integer points in domains
The results
Example: n = 2 and p = 1
Fact
Assume that F is the one-dimensional linear subspace of R2 spanned
by (1, α) ∈ R2 .
For any bounded domain S in R2 with smooth boundary, we get
1. For α 6∈ Q,
nε (S) = ε−1 area(S) + O(ε−1/2 ),
ε → 0.
2. For α = qp , where p and q are coprime,
X
1
nε (S) = ε−1 p
|S ∩ Lk | + O(1),
p2 + q 2 k ∈Z
Yuri A. Kordyukov (Ufa, Russia)
Adiabatic limits and integer points
ε → 0.
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Integer points in domains
Applications to adiabatic limits
The setting
Tn = Rn /Zn the n-dimensional torus.
F a linear foliation on Tn : the leaf Lx of F through x ∈ Tn :
Lx = x + F
mod Zn .
Fact
For any λ ∈ R, we have
nε (B√λ (0)) = Nε (4π 2 λ),
where B√λ (0) is the ball in Rn of radius
Yuri A. Kordyukov (Ufa, Russia)
√
λ centered at the origin.
Adiabatic limits and integer points
Moscow, Delone120, 2010
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Integer points in domains
Applications to adiabatic limits
The main result
Theorem (Yu.K., A. Yakovlev, 2010)
For λ > 0, the following asymptotic formula holds as ε → 0:
Nε (λ) = ε
(n−r )/2
X λ
∗ 2
− |γ |
+ O(εk −q ),
|Q| ∗ ∗ 4π 2
−q ωn−r
γ ∈Γ
where ωn−r is the volume of the unit ball in Rn−r and
( q+1
if q−1
2 ≤p−r
2(p−r +1) ,
k=
2q
q−1
q+1+2(p−r ) , if 2 > p − r .
Yuri A. Kordyukov (Ufa, Russia)
Adiabatic limits and integer points
Moscow, Delone120, 2010
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