Isospectral sets of drumheads are compact in C ∞
Richard Melrose1
Mathematical Sciences Research Institute
and
Massachusetts Institute of Technology
1
Introduction.
The trace of the heat kernel for the Laplacian with Dirichlet boundary condition on C ∞ bounded planar domain, which we call a drumhead following
the euphonius terminology of M. Kac [2], has an asymptotic expansion near
t = 0:
X
1
(1.1)
H(t) = tr (e−t∆ ) ∼ t−1
(ak tk + bk tk+ 2 ) , t ↓ 0 ,
k≥0
with coefficients which are both spectral and geometric invariants of the domain. In this note the form of the bk is examined by perturbation arguments
and it is shown that in terms of the curvature as a function of arclength,
κ(s), with κp (s) = dp κ(s)/ dsp ,
Z
L
2
|κk (s)| ds +
(1.2) bk+1 = ck
0
X
Z
dα
0
α
L
α
k−1
κα0 κα1 1 · · · κk−1
ds ,
k ≥ 3.
Here ck 6= 0 and the sum, which is finite, consists of lower order terms in the
sense that αk−1 ≤ 2. The bk+1 therefore bound the corresponding Sobolev
norms on κ and the compactness, in the C ∞ topology on κ, of the set of
domains with a fixed spectrum follows.
The coefficients a0 , a1 , a2 , a3 , b0 , b1 , b2 have already been computed, see
L. Smith [5] and the earlier results cited there. Similar asymptotic results
1
This work was supported in part by the National Science Foundation under grant
MCS 8006521.
1
have been obtained by S. Marvizi and the author [4] for the wave invariants
introduced in [3]. Further consequences of the formula (1.2) can also be
found in [4].
A similar computation to that made here shows that
(1.3)
Z L
Z L
X
αk−1
ak+2 = Ck
κ(s)|κk (s)|2 ds +
Dα
κα0 κα1 1 · · · κk−1
ds , Ck 6= 0 .
0
0
α
The author would like to thank John Ball for helpful conversations.
2
Heat invariants.
It follows from invariant theory, or a computation of the type described below, that each of the coefficients ak , k ≥ 1, bk , k ≥ 0, in (1.1) can be written
as an integral with respect to arclength of a polynomial in the curvature and
its derivatives:
Z
(2.1)
bk =
L
qk (κ, κ1 , . . . , κp ) ds
k ≥ 0.
0
The polynomials qk are not uniquely determined by (2.1) so it is convenient
to demand the normalization condition:
(2.2)

P
β
qk (x0 , x1 , . . . , xp ) =
β∈B cβ x with cβ 6= 0 , ∀ β ∈ B and if
P (β) = max (β 6= 0) then β ∈ B , p = P (β) ≥ 1 implies that β ≥ 2 .
p p
p
That is, in each term of qk the highest derivative occurs at least quadratically. The existence of qk satisfying (2.1), (2.2) is easily established by
integration by parts; its uniqueness can also be shown but this will not be
needed here.
There is a natural transformation which can be used to deduce a homogeneity property of the bk . Namely, consider radial expansion from an
interior point of the domain by a factor R. Under such expansion
(2.3)
L0 = RL ,
κ0 (s) = R−1 κ(r−1 s) .
Moreover, the eigenvalues of the Dirichlet problem are transformed according to λ0k = R2 λk . Thus,
X
(2.4)
H 0 (t) =
exp(−tλ0k ) = H(R−2 t) .
k
2
This shows that the invariants have the transformation law
b0k = R1−2k bk
(2.5)
under the transformation (2.3). Assigning to a monomial xβ the weight
(2.6)
W (β) =
p
X
(j + 1)Bj
j=0
it follows that (2.1) continues to hold if qk is replaced by the sum of its terms
of weight W (β) = 2k.
From this it follows immediately that
L k 2
d κ
Z
(2.7)
bk+1 = ck
dsk
0
L
Z
ds +
qk0 (κ, κ1 . . . κk−1 ) ds
0
where the lower order terms are at least cubic, but at most quadratic in
κk−1 for k ≥ 3,
(2.8) qk0 (κ, κ1 , . . . , κk−1 ) = qk00 (κ, κ1 , . . . , κk−2 )κ2k−1 + qk00 (κ, κ1 , . . . , κk−2 ) ,
with all terms of weight 2k + 2. The main result of this note is:
Proposition 1. In the representation (2.7) of bk+1 the coefficient ck is not
zero.
This is proved in Section 4 below. As a simple consequence we obtain:
Theorem 1. Each set I(Ω) of C ∞ bounded domains with the same Dirichlet
spectrum as a given domain Ω is (arc-)compact in the C ∞ topology on the
curvature.
Proof. First note that the length L of the boundary is constant on I = I(Ω),
RL
as is the number of boundary components. Similarly, b1 = c1 0 κ2 ds so
from the constancy of b1 on I it follows that I is bounded in L2 , as a norm
on the curvature. Next consider
Z L
Z L
(2.9)
b2 = c2
κ21 ds + c02
κ4 ds ,
0
0
using (2.2). The Sobolev embedding theorem shows that for any δ > 0,
!
3 Z
Z
Z
Z
L
L
κ4 ds ≤ δ
0
L
L
κ2 ds
κ21 ds + cδ
0
0
3
κ2 ds
+
0
.
R
Thus from (2.9) and the constancy of b2 on I it follows that κ21 ds is
uniformly bounded on I. This in particular implies the boundedness of the
supremum norm kκk∞ . Next consider
Z L
Z L
Z L
2
0
3
(2.10)
b3 = c3
κ2 ds + c3
κ1 ds +
q200 (κ, κ1 ) ds ,
0
0
0
using (2.2), with q200 at most quadratic in κ1 . Thus the third term in (2.10)
has already been shown to be bounded, and a similar argument to that above
shows that the second term is also bounded in terms of b0 , b1 , b2 ; thus the
uniform boundedness of the L2 norm kκ2 k2 follows from the boundedness
of b3 .
Proceeding inductively we can suppose that for each p < k, k > 2, the
boundedness of b0 , b1 , . . . bp+1 implies the boundedness of kκr k2 , r ≤ p − 1
and kκp k2 . From (2.7) and (2.8) these estimates imply the boundedness of
the second term in (2.7) and hence the boundedness of kκk k2 , and so of
kκp k∞ for p ≤ k − 1, given the boundedness of br , 0 ≤ r ≤ k + 1. This
completes the proof, by induction, that the set I(Ω) is bounded in C ∞ .
3
Perturbation formulae.
The heat kernel associated to the Laplace operator with Dirichlet boundary
condition can be computed in terms of the Schwartz kernel, r(λ, z, z 0 ), of
the resolvent through
Z
0
−1
(3.1)
h(t, z, z ) = (2πi)
e−λt r(λ, z, z 0 ) dλ
z, z 0 ∈ Ω, t > 0
Γ
where Γ is a contour in C/R̄+ tending to infinity in the negative direction in
Re(λ) > 0, Im(λ) > 0 and in the positive direction in Re(λ) > 0, Im(λ) < 0.
The kernel r satisfies
(
(∆z − λ)r(λ, z, z 0 ) = δ(z, z 0 ) in Ω × Ω
(3.2)
r(λ, z, z 0 ) = 0 z ∈ ∂Ω , z 0 ∈ Ω .
From the standard properties of the kernels of pseudodifferential operators, and Poisson operators (see Boutet de Monvel [1]), R(λ), with kernel
r(λ, z, z 0 ), is not trace class but ∂λ R(λ), with kernel ∂λ r(λ, z, z 0 ) is trace
class having continuous kernel. Thus,
Z Z
−1
(3.3)
H(t) = (2πit)
e−λt ∂λ r(λ, z, z) dz dλ , t > 0 .
Γ
4
Provided Γ lies outside a conic neighborhood of R̄+ , the trace
Z
(3.4)
T (λ) = ∂λ r(λ, z, z) dz
is a classical symbol as |λ| → ∞, along Γ, and computing the fractional
powers in the expansion of H(t) can be accomplished by computing the
appropriate terms in the expansion of T (λ).
Now, suppose that the boundary ∂Ω of Ω undergoes a smooth deformation giving the one parameter family of domains Ω . The theory of elliptic
boundary problems shows that
Z
(3.5)
T (λ) =
∂λ r (λ, z, z) dz
Ω
is C ∞ in . To compute the coefficient ck in (2.7) it is enough to be able
to find the expansion, to all orders, of d2 T (λ)/ d2 at = 0, around any
convenient domain in terms of the curvature variation.
To simplify this computation we shall choose as base domain the halfspace y ≥ 0, i.e. Z = Rx × R+
y . Of course, this is not compact but variations
satisfy the same formula, as we shall see. Thus, consider the solution of

0
0

(∆ − λ)r (λ, z, z ) = δ(z − z ) in Z × Z
(3.6)
r (λ, x, τ (x)z 0 ) = 0


r (λ, z, z 0 ) → 0 as |z| → ∞ .
where Z = {y > τ (x)} and the boundary perturbation, y = τ (x) has
compact support. Now, the arclength differential on the boundary is
ds = (1 + 2 (τ 0 (x))2 )1/2 dx
(3.7)
and the curvature is
x (x) = τ 00 (x)(1 + 2 (τ 0 (x))2 )−1/2 .
(3.8)
Thus, at = 0, the second variation is
(3.9)
∂2
Z
∞
(κk (s))2 ds
Z
=2
0
0
∞
2
dk+2
τ (x)
dx .
dxk+2
If r is defined by (3.6) then set
(3.10)
r = r0 (λ, z, z 0 ) ,
r=
d
r0 (λ, z, z 0 ) ,
d
5
r00 =
d2
r0 (λ, z, z 0 )
d2
all kernels on Z × Z, and consider the symbol, outside a cone around R̄+
in C
Z ∞Z ∞
T2 (λ) =
∂λ r00 (λ, x, y ; x, y) dy dx
−∞ 0
Z ∞
(3.11)
−
2τ (x)∂λ r0 (λ, x, 0 ; x, 0) dx
Z−∞
∞
+
τ 2 (x)(∂y + ∂y0 )∂y r(λ, x, 0 ; x, 0) dx .
−∞
Proposition 2. T2 (λ) has a classical expansion as |λ| → ∞ outside any
cone containing R̄+ , of the form
(3.12)
T2 (λ) ∼
X
Z
∞
ep (λ)
−∞
p≥1
dp τ
dxp
2
dx + f (λ)τ (x) dx
where the order of ep tends to −∞ as p → ∞. The coefficients ep (λ) are,
modulo S −∞ , the same as would be obtained by expansion of d2 T (λ)/d2 ,
at = 0, for the same perturbation y = τ (x) of the boundary y = 0 of the
cylindrical domain Rx /pZ × [0, 1] for p so large that supp(τ ) ⊂ − 12 p , 21 p .
Proof. Consider the compact manifold, with flat metric,
X = Rx /pZ × [0, 1]
under perturbation to X , with the boundary y = 0 modified to y = τ (x),
− 21 p ≤ x ≤ 12 p being considered as a fundamental domain for R/pZ. The
existence of an expansion for T (λ), given by (3.5) is standard, with the
coefficients again being the invariants ak , bk of (1.1). The existence of an
expansion (3.12) for d2 T 0 (λ)/d2 therefore follows from formulae (3.7), (3.8)
but now for perturbation X of X. Only terms linear or quadratic in τ can
occur, so the expansion must be of the form (3.12). Moreover, if r̄ (λ, z, z 0 )
is the corresponding kernel, satisfying
(
(∆ − λ)r̄ (λ, z ; z 0 ) = δ(z − z 0 )X × X
(3.13)
r̄ (λ, z, z 0 ) = 0 y = τ (x) on y = 1
then from (3.5),
(3.14)
Z
T (λ) =
1
p
2
− 12 p
Z
1
∂λ r̄ (λ, x, y ; x, y) dy dx .
τ (x)
6
Direct differentiation gives:
Z
Z
2 0
2
00
(3.15) d T /d =
∂λ r̄ (λ, z, z) dz − 2
− 12 p
X
Z
−
1
p
2
1
p
2
− 12 p
τ (x)∂λ r̄0 (λ, x, 0 ; x, 0) dx
τ 2 (x)(∂y + ∂y0 )∂λ r̄(λ, x, 0 ; x, 0) dx
where r̄, r̄0 , r̄00 are defined as in (3.10) but from r̄ .
Finally, to prove (3.12) it is enough to show that the difference between
each term in (3.11) and the corresponding term in (3.15) is rapidly decreasing
as |λ| → ∞, outside a cone containing R̄+ . Consider first the passage from
the kernel r satisfying (3.6) to the corresponding kernel r̃ where Rx is
replaced by
Rx /pZ, assuming always that the perturbation has support in
− 12 p , 12 p . In fact,
X
r̃ (λ, z, z 0 ) =
r (λ, z + kp, z 0 ) .
k∈Z
Standard elliptic estimates show that on X the difference
(3.16)
∂ r̃ − ∂ r = 0 |λ|−N ∀ N .
Similarly, in passing from r̃ to r̄ a correction term must be added to give
the Dirichlet condition at y = 1, rather than decrease at infinity. Since
the boundary condition is independent of , and at a finite distance from
the perturbed boundary, the difference between each term in (3.11) and the
corresponding term in (3.15) is rapidly decreasing as |λ| → ∞, along Γ. This
completes the proof of Proposition 2.
We shall now proceed to derive the expansion (3.12) explicitly. The point
of the proposition is that, by comparison with the expansion of d2 T 0 / d2 ,
the coefficients can be related to the ck in (2.7).
4
Computation of coefficients.
Starting from the formula (3.11) we shall compute the form of the expansion (3.12). Thus, it is first necessary to find r, r0 , r00 and associated kernels.
If (ξ 2 − λ)1/2 is for ξ ∈ R, λ ∈ R̄+ the branch with positive real part, then
(4.1)
(
R
0
2
1/2
0
(4π)−1 ei(x−x )ξ−(ξ −λ) (y−y ) (2 − λ)−1/2 dξ , y > y 0
0
R
F (λ, z, z ) =
0
2
1/2 0
(4π)−1 ei(x−x )ξ−(ξ −λ) (y −y) (2 − λ)−1/2 dξ , y 0 > y
7
is the preferred, i.e., temperate, fundamental solution of ∆ − λ on R2 . In
terms of this kernel,
(4.2) r(λ, z, z 0 ) = F (λ, z, z 0 ) − G(λ, z, z 0 )
Z
0
2
1/2
0
(4.3) G(λ, z, z 0 ) = (4π)−1 ei(x−x )ξ−( −λ) (y+y ) (ξ 2 − λ)−1/2 dξ .
Thus, by continuity
(4.4)
(∂y + ∂y0 )∂λ r(λ, x, 0 ; x, 0) = 2∂y ∂y G(λ, x, 0 ; x, 0) = 0 ,
so the third term in (3.11) is identically zero.
The corresponding Poisson kernel, solving
(
(∆ − λ)P (λ, z ; x0 ) = 0
(4.5)
P (λ, x, 0 ; x0 ) = δ(x − x0 )
and vanishing at infinity is
(4.6)
P (λ, z ; x0 ) = (2π)−1
Z
0
ei(x−x )ξ−(ξ
2 −λ)1/2 y
dξ .
Differentiating (3.6) gives the boundary problem for r0 :
(4.7)
(∆ − λ)r0 (λ, z ; z 0 ) = 0 ,
r0 (λ, x, 0 ; z 0 ) = −τ (x)∂y r(λ, x, 0 ; z 0 ) ,
with r0 vanishing at infinity. Thus, from (4.6) and (4.2),
Z
0
00
0
2
1/2
0 2 1/2 0
0
0
−2
(4.8)r (λ, z ; z ) = lim −(2π)
ei(x−x )ξ+i(x −x)ξ −(ξ −λ) y−((ξ ) ) y
χ→1
·τ (x00 )χ(ξ)χ(ξ 0 ) dξ 0 dξ dx00 .
Where compactifying cutoffs χ have been inserted to ensure absolute convergence, and the limit is taken over χ → 1 in the symbol topology S δ , δ > 0.
Differentiating (3.6) twice gives the corresponding boundary problem for r00 :
(
(∆ − λ)r00 (λ, z ; z 0 ) = 0
(4.9)
r00 (λ, x, 0 ; z 0 ) = −2τ (x)∂y r0 (λ, x, 0 ; z 0 ) − τ 2 (x)∂y2 r(λ, x, 0 ; z 0 ) .
From (4.2), ∂y2 r = 0, at y = 0, z 0 ∈ Z so the second term in the boundary
condition for r00 is zero. Applying (4.6) again allows one to construct r00 , we
only need
Z
0
0
00 0
00
00
(4.10)
∂λ r00 (λ, z, z) = lim ∂λ − 2(2π)−3 ei(x−x )ξ+i(x −x )ξ +i(x −x)ξ
χ→1
−(ξ 2 −λ)1/2 y−((ξ 00 )2 −λ)1/2 y
·e
((ξ 0 )2 − λ)1/2
·τ (x0 )τ (x00 )χ(ξ)χ(ξ 0 )χ(ξ 00 ) dξ 00 dξ 0 dξ dx00 dx0 .
8
Again from (4.8), the second term in (3.11) is identically zero, so
Z
0
0
00 0
00
00
(4.11) T2 (λ) = lim ∂λ − 2(2π)−3 ei(x−x )ξ+i(x −x )ξ +i(x −x)ξ
χ→1
0
00
·τ (x )τ (x )χ(ξ)χ(ξ 0 )χ(ξ 00 )((ξ 0 )2 − λ)1/2 ((ξ 2 − λ)1/2 + ((ξ 00 )2 − λ)1/2 )−1
· dξ dξ 0 dξ 00 dx0 dx00 dx .
In (4.11) the integral over x, ξ 00 can be evaluated directly, since the integrand
is independent of x, giving:
Z
0
0
−2
(4.12) T2 (λ) = lim ∂λ − (2π)
ei(x −x)(ξ −ξ) τ (x0 )τ (x)
λ→ξ
·χ(ξ)χ(ξ 0 )((ξ 0 )2 − λ)1/2 (ξ 2 − λ)−1/2 dξ dξ 0 dx0 dx .
The integrals over x0 and ξ 0 can now be evaluated by the lemma of stationary
phase:
Z
X
(4.13) T2 (λ) ∼
ek (λ) (dk τ (x)/ dxk )2 dx
k≥1
ek (λ) = −((2k)!(2π))−1 ∂λ
Z
(ξ 2 − λ)−1/2 ∂ξ2k (ξ 2 − λ)1/2 dξ
the odd order terms being automatically zero.
The coefficient ek (λ) is actually homogeneous of degree −k − 12 , so from
its holomorphy properties
1
ek (λ) = γk−2 (−λ)−k− 2 ,
(4.14)
with
(4.15)
γk−2
Z
1
−1
=− k−
((2k)!(2π))
(s2 + 1)−1/2 d2k (s2 + 1)1/2 / ds2k ds
2
Z
1 2
k
−1
= (−1) k −
((2k)!2π)
(dk−1 (s2 + 1)−1/2 / dsk−1 )2 ds .
2
This shows that all the coefficients γk−2 , k ≥ 2, are non-zero. Tracing
backwards through (3.12) it follows easily that all the ck , k ≥ 0, are nonzero, providing Proposition 1.
More explicitly a short computation gives the formula
Z
1
1
(4.16)
ck = γk
e−λ(−λ) −k−5/2 dλ
k > 0,
2 2π Γ
9
using (4.15) this becomes:
(4.17)
2
2
n
ck = Γ(−k − 3/2)(k + 3/2) (4π · (2k + 4)!)
Z
(dk+1 (1 + s2 )−1/2 / dsk+1 ) ds .
References
[1]
L. Boutet de Monvel, “Boundary problems for pseudo-differential operators”, Acta. Math. 126 (1971), 11–51.
[2]
M. Kac, “Can one hear the shape of the drum?”, Amer. Math. Monthly
73 (1966), 1–23.
[3]
S. Marvizi and R.B. Melrose, “Spectral invariants of convex planar
regions”, J. Diff. Geom. 17 (1982), 475–502.
[4]
S. Marvizi and R.B. Melrose, “Wave invariants and spectral determinacy”. In preparation.
[5]
L. Smith, “The asymptotics of the heat equation for a boundary value
problem”, Invent. Math. 63 (1981), 467–493.
10