CBMS Lecture: Local Weyl laws
Steve Zelditch
Northwestern
Kentucky
June 20, 2011
Local Weyl law asymptotics
The classical Weyl law asymptotically counts the number of
eigenvalues less than λ,
N(λ) = #{j : λj ≤ λ} =
|Bn |
Vol(M, g )λn + O(λn−1 ).
(2π)n
(1)
Here, |Bn | is the Euclidean volume of the unit ball and Vol(M, g )
is the volume of M with respect to the metric g . Equivalently,
TrEλ =
Vol(|ξ|g ≤ λ)
+ O(λn−1 ),
(2π)n
(2)
where Vol is the P
symplectic volume measure relative to the natural
symplectic form nj=1 dxj ∧ dξj on T ∗ M. Thus, the dimension of
√
the space where H = ∆ is ≤ λ is asymptotically the volume
where its symbol |ξ|g ≤ λ.
When periodic geodesics form a set of measure zero
We say that (M, g ) is aperiodic if the periodic geodesics form a set
of measure zero in S ∗ M. In this case the remainder estimate has
been improved by Duistermaat-Guillemin-Ivrii and one can obtain
Weyl laws over the shorter interval [λ, λ + 1].
We also say that (M, g ) is periodic if the set of closed geodesics
has positive measure. For instance, the round sphere and Zoll
manifolds are periodic.
Generic g are aperiodic in this sense.
Two term Weyl laws
An improved, two-term Weyl law has
√ been proved which takes into
account the singularities of Tr cos t ∆ for larger values of t. The
singular t 6= 0 are the lengths of the closed geodesics γ of g t . The
size of the remainder reflects the measure of closed geodesics.
1. In the aperiodic case, Duistermaat-Guilleimin (-Ivrii’)s two
term Weyl law states
N(λ) = #{j : λj ≤ λ} = cm Vol(M, g ) λm + o(λm−1 )
where m = dim M and where cm is a universal constant.
√
2. In the periodic case, the spectrum of ∆ is a union of
eigenvalue clusters CN of the form
CN = {(
2π
β
)(N + ) + µNi , i = 1 . . . dN }
T
4
with µNi = 0(N −1 ). The number dN of eigenvalues in CN is a
polynomial of degree m − 1.
Local Weyl laws
One of the principal methods for relating eigenfunctions and
geodesic flow are the local Weyl laws. One form of the local Weyl
law is to find the asymptotics and remainder for the pointwise
sums,
X
N(λ; x) =
|ϕλj (x)|2 .
j:λj ≤λ
Another is to find, for A ∈ Ψ0 , the asymptotics and remainder for
X
NA (λ) =
hAϕλj , ϕλj i.
j:λj ≤λ
Uniform pointwise local Weyl laws
The first local Weyl law is the un-integrated version of Weyl’s law
for counting eigenvalues:
Theorem
(Avakumovich, Levitan, Hörmander, Duistermaat-Guillemin)
X
λj ≤λ
|ϕj (x)|2 =
1
|B n |λn + R(λ, x),
(2π)n
(3)
where R(λ, x) = O(λn−1 ) uniformly in x. Here, |Bn | is the volume
of the unit ball in Rn .
The size of the remainder depends on the structure of geodesic
loops at x. This has been studied in detail by Safarov et al,
Sogge-Z, Ivrii and others.
Improvement if the loops Lx are set of measure zero
The remainder is sharp on the round sphere or in general if all
geodesics from x loop back to x. Safarov et al (and Sogge-Z)
observed that one can generalize the Duistermaat-Guillemin (-Ivrii)
Weyl law if the set of loops has zero measure at every x.
Theorem
If the measure in Sx∗ M of the set of geodesic loops Lx at x has
measure zero for all x in Sx∗ M, then
X
|ϕj (x)|2 =
λj ≤λ
where R(λ, x) = o(λn−1 ).
1
|B n |λn + R(λ, x),
(2π)n
Local Weyl law for PsiDO’s
The PsiDO local Weyl law concerns the traces TrAE √
(λ) where
A ∈ Ψm (M). Here E (λ) is the spectral projector for ∆ for the
spectral interval [0, λ]. We often write the projector as Π[0,λ] .
Theorem
X
λj ≤λ
1
hAϕj , ϕj i =
(2π)n
Z
B ∗M
σA dxdξ λn + O(λn−1 ).
(4)
Fourier Tauberian approach
In both cases, the asymptotics are determined by a Fourier
Tauberian method, by studying the singularities of the dual trace,
X
S(t, x) =
e itλj |ϕλj (x)|2 ,
j
resp.
SA (t) =
X
e itλj hAϕλj , ϕλj i
j
= TrAe it
√
∆
.
√
We could as (or more) easily use cos t ∆.
Singularities and smoothed Weyl sums
The idea of cosine or Fourier Tauberian theory is to determine
spectral asymptotics from the singuarity at t = 0 of the pointwise
value
X
U(t, x, x) =
e itλj ϕλj (x)2
j
√
on the diagonal. Here, we could also use cos t ∆.
By the standard techniques of wave front sets, it is the same to fix
a test function ρ ∈ S(R) with supp ρ̂ contained in a sufficiently
small neighborhood of 0 and ρ̂ ≡ 1 in a small neighborhood of 0,
and to study the asymptotics of the smoothed Weyl sums,
X
ρ(λ − λj )|ϕλj (x)|2
j
as λ → ∞
Duistermaat-Guillemin short time asymptotic expansion
Proposition
Let (M, g ) be a C ∞ compact Riemannian manifold of dimension
n. Then there exists a sequence ω1 , ω2 , . . . of real valued smooth
densities on M such that, for every ρ ∈ S(R) with supp ρ̂
contained in a sufficiently small neighborhood of 0 and ρ̂ ≡ 1 in a
small neighborhood of 0,
X
j
2
ρ(λ − λj )|ϕλj (x)| ∼
∞
X
ωk λn−k−1
k=0
as λ → ∞ (and rapidly decaying as λ → −∞) with
ω0 (x) = Vol(Sx∗ M), ω1 = 0 = ωn ; ωk = 0 for odd k.
Sketch of proof
One uses a short-time parametrix,
Z
−n
U(t, x, y ) = (2π)
e iϕ(t,x,y ,η) a(t, x, y , η)dη
Rn
where a is a classical symbol of order m and where
ϕ(t, x, y , η) = ψ(x, y , η) − t|η|, with ψ(x, y , η) = 0 if
hx − y , ηi = 0. Hence,
Z
ρ ∗ dN(λ, x) =
e iλt ρ̂(t) U(t, x, x)dt
R
Z Z
−n
e iλt ρ̂(t)e −it|η| a(t, x, x, η) dη dt .
= (2π)
R
Rn
One now changes variables η → λη, puts the dξ integral into polar
coordinates ξ = r ω, |ω| = 1 and carries out the dtdr integral by
the method of stationary phase.
Local Weyl law
The same kind of argument applies to NA (λ):
Proposition
P
For A ∈ Ψm (M), let NA (λ) = λj ≤λ (Aϕj , ϕj ). Then for any
ρ ∈ S(R) with ρ̂ ∈ C0∞ (R), supp ρ̂ ∩ Lsp(M, g ) = {0} and with
ρ̂ ≡ 1 in some interval around 0, we have:
ρ ∗ dNA (λ) ∼
∞
X
αk λn+m−k−1 (λ → +∞)
k=0
where:
Z
n = dim M, α0 =
S ∗M
Z
σA dµ, αk =
S ∗M
ωk dµ
where ωk is determined from the k-jet of the complete symbol a of
A.
Sketch of proof
Proof. The only new step is to apply A to the parametrix for Ut .
Applying a PsiDO to ae iϕ produces an expression αe iϕ with the
same phase and only a change in the amplitude. Hence,
Z
−n
α(t, x, y , η)e iϕ(t,x,y ,η) dη
AU(t, x, y ) = (2π)
Rn
where α is a classical symbol of order m. Now proceed as before.
Action of a pseudo-differential operator on a rapidly
oscillating exponential
One of the key calculations is the action of P(x, D) on a rapidly
oscillating WKB Lagrangian state ae iτ ϕ .
Proposition
Let ϕ be real-valued and suppose that dϕ 6= 0 (anywhere). Then
P(x, D)(a(x, ξ)e iτ ϕ(x,ξ) ) = aP (x, ξ)e iτ ϕ(x,ξ)
with
aP (x, ξ) ∼
X 1
D α p(x, τ dϕ)Nα (ϕ, τ, Dx )a(x, ξ),
α!
α
where
Nα (ϕ, τ, Dx )u(x) = Dyα e iτ ϕ(2) (y ,ξ) u(y )|y =x ,
ϕ(2) being the second order remainder of the Taylor expansion of
ϕ(y , ξ) around y = x: ϕ(x) − ϕ(y ) = ∇ϕx · (x − y ) + ϕ(2) (x, y ).
Long time pre-trace formulae
We now discuss pre-trace formulae for U(t, x, x) for long times t.
The main difficulty is that there may not exist a better parametrix
for U(t, x, x) than to use the group formula U(t) = U(t/N)N to
reduce to times < inj(M, g ). But this will require N × m integral
signs and amplitudes that are difficult to control. Moreover, the
singularities become very difficult to control.
Manifolds without conjugate points
One can get a good idea of the difficulties by studying the simplest
case, that of manifolds without conjugate points. This case is a
rather straightforward generalization of the Selberg pre-trace
formula for compact hyperbolic manifolds. The key feature is that
there exists a global in time parametrix on the universal cover, so it
is not necessary to assume that the suppport of ρ̂ is small. As in
the classical trace formula, one organizes the elements γ ∈ Γ (the
deck transformation group) into conjugacy classes γ̂. One then has
X
U(t, x, x) =
Uγ̂ (t, x, x),
γ̂
where
Uγ̂ (t, x, y ) =
X
α∈Γ/Γγ
Ũ(t, x, α−1 γαx) =
X
Ũ(t, αx, γαx),
α∈Γ/Γγ
since α is an isometry. Here, Γγ is the stabilizer in Γ of γ.
Hadamard parametrix
We the Hadamard parametrix . Since we are interested in the long
time singularities, we can use the phase function r − t instead of
r 2 − t 2 . Either one parameterizes the graph of the geodesic flow
away from r = 0. One then has,
XZ
ρ ∗ dN(λ, x) =
e iλt ρ̂(t) Ũ(t, x, γx)dt
γ∈Γ
=
X X Z
R
γ̂ α∈Γ/Γγ
=
X X Z
γ̂ α∈Γ/Γγ
Z
R
R
e iλt ρ̂(t)e iθ(r (αx,γαx)−t)) a(αx, γαx, θ) dθ dt
R+
Z
R+
ρ̂(t)e iλ(t+θ(r (αx,γαx)−t)) a(αx, γαx, θ) dθ dt .
Stationary phase analysis
To determine the asymptotics as λ → ∞ one again applies
stationary phase. The phase function is critical when
θ = 1, t = r (αx, γαx). This corresponds to the times t when there
exists a geodesic loop at x ∈ M (i.e. at αx ∈ M̃). The geodesic
loops are in one-one correspondence with conjugacy classes in Γ
and hence form a countable set. However, the growth rate of this
set as t → ∞ is often exponentially large. Thus, if supp
ρ̂ ⊂ [−T , T ], then there are often e CT terms in the sum over γ.
This forces us to restrict to times T ≤ C log λ.
With a specific C , this is known as the Ehrenfest time.
Semi-classical asymptotics often break down at the Ehrenfest time.
Pointwise Weyl law with remainder
Theorem
(Selberg; Berard) Let (M, g ) be compact and without conjugate
points. Then
λn−1
N(λ, x) = |B|n λn + O(
).
log λ
Idea of proof: The leading term comes from the identity term
γ = id in the sum over γ. The other terms have weaker
singularities. But the number of γ of length ≤ T grows like e hT (h
= topological entropy). This forces one to choose a test function
supported in [0, C log λ].
Safarov trace formula
Safarov added some precision to the Duistermaat-Guillemin
singularities pre-trace formula. For fixed x, Given x ∈ M, we let Lx
denote the set of loop directions at x:
Lx = {ξ ∈ Sx∗ M : ∃T : expx T ξ = x}.
(5)
We let Tx : Sx∗ M → R+ ∪ {∞} denote the return time function to
x,

 inf{t > 0 : expx tξ = x}, if ξ ∈ Lx ;
Tx (ξ) =

+∞,
if no such t exists.
We then define the first return map by Φx = gxTx : Lx → Sx∗ M. We
also define T (k) (ξ) to be the time of kth return for directions
which loop back at least k times.
Pointwise Perron-Frobenius operator
We then consider the positive partially unitary operator (the
Perron-Frobenius operator)
Ux : L2 (Lx , |dω|) → L2 (Sx∗ , |dω|),
p

 f ((Φx )(ξ)) Jx (ξ), ξ ∈ Lx ,
Ux f (ξ) =

0,
ξ∈
/ Lx .
Here, Jx is the Jacobian of the map Φx , i.e. Φ∗x |dξ| = Jx (ξ)|dξ|.
Safarov trace formula
We have:
ker Ux = {f ∈ L2 (Sx∗ ) : supp f ∩ Φx (Lx ) = ∅};
=Ux = {f ∈ L2 (Sx∗ ) : supp f ⊂ Lx }.
We further define
±
Ux± (λ) = e iλTx Ux± .
Safarov trace formula
τ
Let ρT be the dilated test function satisfying ρc
T (τ ) = ρ̂( T ). The
pre-trace formula then has the form,
ρT ∗ dN(λ, x) = a0 (x)λn−1
+λn−1
Z
∞
X
Lx k=1
(k)
ρ̂(
Tx (ξ)
)Ux (λ)k |dξ| + oT ,x (λn−1 ).
T
A key point in the proof is that the phase of the oscillatory integral
for the left side only has a stationary phase point at ξ ∈ Lx . That
reduces the integral to one over Lx modulo an error oT (λn−1 ).
Local Weyl laws for FIO’s
One may study the same Weyl sums when A is a Fourier integral
operator.
Proposition
(Z, 1986) Let CF ⊂ T ∗ M − 0 × T ∗ M − 0 be a local canonical
graph and F ∈ I 0 (M × M; CF ). Then,
Z
X
1
lim
hF ϕj , ϕj i =
σF dµL .
λ→∞ N(λ)
S(CF ∩∆S ∗ M×S ∗ M )
j:λj ≤λ
Here, S(CF ∩ ∆S ∗ M×S ∗ M ) is the set of unit vectors in the diagonal
part of CF .
Idea of proof
We again take a Fourier Tauberian approach and now study the
singularities of
√
Tr F e it ∆
at t = 0. Using the composition calculus of FIO’s the singularity is
determined by the intersection of the canonical relation of F with
the diagonal, i.e. with the fixed point set of the canonical relation.
The integral is with respect to a canonical volume form on this
fixed point set coming from the symbol calculus. The limit sifts
out the pseudo-differential part (i.e. the diagonal branch) of the
canonical relation.
If the fixed point set has measure zero, then the limit is zero. One
can then amplify F by composing it with ∆s for some s.
Tauberian theorems
We record here the statements of the Tauberian theorems that we
use in the article. Our main reference is Safarov-Vasileev Appendix
B and we follow their notation.
We denote by F+ the class of real-valued, monotone nondecreasing
functions N(λ) of polynomial growth supported on R+ . The
following Tauberian theorem uses only the singularity at t = 0 of
c to obtain a one term asymptotic of N(λ) as λ → ∞:
dN
Theorem
Let N ∈ F+ and let ψ ∈ S(R) satisfy the conditions: ψ is even,
ψ(λ) > 0 for all λ ∈ R, ψ̂ ∈ C0∞ , and ψ̂(0) = 1. Then,
ψ ∗ dN(λ) ≤ Aλν =⇒ |N(λ) − N ∗ ψ(λ)| ≤ CAλν ,
where C is independent of A, λ.
Two term Tauberian theorems
To obtain a two-term asymptotic formula, one needs to take into
c We let ψ be as above, and
account the other singularities of dN.
also introduce a second test function γ ∈ S with γ̂ ∈ C0∞ and with
the supp γ̂ ⊂ (0, ∞).
Theorem
Let N1 , N2 ∈ F+ and assume:
1. Nj ∗ ψ(λ) = O(λν ), (j = 1, 2);
2. N2 ∗ ψ(λ) = N1 ∗ ψ(λ) + o(λν );
3. γ ∗ dN2 (λ) = γ ∗ dN1 (λ) + o(λν .
Then,
N1 (λ − o(1)) − o(λν ) ≤ N2 (λ) ≤ N1 (λ + o(1)) + o(λν ).
Hörmander Tauberian theorem
Lemma
Suppose that µ is a non-decreasing temperate function satisfying
µ(0) = 0 and that ν is a function of locally bounded variation such
that ν(0) = 0. Suppose also thatRm ≥ 1 and that ϕ ∈ S(R) is a
fixed positive function satisfying ϕ(λ)dλ = 1 and ϕ̂(t) = 0,
t∈
/ [−1, 1]. If ϕσ (λ) = σ −1 ϕ(λ/σ), 0 < σ ≤ σ0 , assume that for
λ∈R
|dν(λ)| ≤ A0 (1 + |λ|)m + A1 (1 + |λ|)m−1 dλ,
(6)
and that
|((dµ − dν) ∗ ϕσ )(λ)| ≤ B(1 + |λ|)−2 .
(7)
Then
|µ(λ) − ν(λ)| ≤ Cm A0 σ(1 + |λ|)m + A1 σ(1 + |λ|)m−1 + B , (8)
where Cm is a uniform constant depending only on σ0 and our
m ≥ 1.