Isospectral sets of drumheads are compact in C^{∞}

Richard Melrose^{1}

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:

(1.1)  H(t) = tr (e^{−tΔ}) ∼ t^{−1} ∑_{k≥0}(a_{k}t^{k} + b_{k}t^{k+ \frac{1}{2}} ) , t ↓ 0 ,

with coefficients which are both spectral and geometric invariants of the domain. In this note the form of the b_{k} is examined by perturbation arguments and it is shown that in terms of the curvature as a function of arclength, κ(s), with κ_{p}(s) = d^{p}κ(s)/ ds^{p},

(1.2) b_{k+1} = c_{k} ∫_{0}^{L} |κ_{k}(s)|^{2} ds + ∑_{α} d_{α} ∫_{0}^{L} κ^{α_{0}}κ_{1}^{α_{1}} ⋯ κ_{k−1}^{α_{k−1}} ds , k ≥ 3 .

Here c_{k} ≠ 0 and the sum, which is finite, consists of lower order terms in the sense that α_{k−1} ≤ 2. The b_{k+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 a_{0}, a_{1}, a_{2}, a_{3}, b_{0}, b_{1}, b_{2} have already been computed, see L. Smith [5] and the earlier results cited there. Similar asymptotic results 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].

1This work was supported in part by the National Science Foundation under grant MCS 8006521.

A similar computation to that made here shows that (1.3)

a_{k+2} = C_{k} ∫_{0}^{L} κ(s)|κ_{k}(s)|^{2} ds + ∑_{α} D_{α} ∫_{0}^{L} κ^{α_{0}}κ_{1}^{α_{1}} ⋯ κ_{k−1}^{α_{k−1}} ds , C_{k} ≠ 0 .

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 a_{k} , k ≥ 1, b_{k} , k ≥ 0, in (1.1) can be written as an integral with respect to arclength of a polynomial in the curvature and its derivatives:

(2.1)  b_{k} = ∫_{0}^{L} q_{k}(κ, κ_{1}, …, κ_{p}) ds k ≥ 0 .

The polynomials q_{k} are not uniquely determined by (2.1) so it is convenient to demand the normalization condition: (2.2)

q_{k}(x_{0}, x_{1}, …, x_{p}) = ∑_{β∈B} c_{β}x^{β} with c_{β} ≠ 0 , ∀ β ∈ B and if

P (β) = max_{p}(β_{p} ≠ 0) then β ∈ B , p = P (β) ≥ 1 implies that β_{p} ≥ 2 .

That is, in each term of q_{k} the highest derivative occurs at least quadratically. The existence of q_{k} 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 b_{k}. Namely, consider radial expansion from an interior point of the domain by a factor R. Under such expansion

(2.3)  L′ = RL , κ′(s) = R^{−1}κ(r^{−1}s) .

Moreover, the eigenvalues of the Dirichlet problem are transformed according to λ′_{k} = R^{2}λ_{k}. Thus,

(2.4)  H′(t) = ∑_{k} exp(−tλ′_{k}) = H(R^{−2}t) .

This shows that the invariants have the transformation law

(2.5)  b′_{k} = R^{1−2k}b_{k}

under the transformation (2.3). Assigning to a monomial x^{β} the weight

(2.6)  W (β) = ∑_{j=0}^{p}(j + 1)B_{j}

it follows that (2.1) continues to hold if q_{k} is replaced by the sum of its terms of weight W (β) = 2k.

From this it follows immediately that

(2.7)  b_{k+1} = c_{k} ∫_{0}^{L} (\frac{d^{k}κ}{ds^{k}})^{2} ds + ∫_{0}^{L} q′_{k}(κ, κ_{1} … κ_{k−1}) ds

where the lower order terms are at least cubic, but at most quadratic in κ_{k−1} for k ≥ 3,

(2.8) q′_{k}(κ, κ_{1}, …, κ_{k−1}) = q′′_{k}(κ, κ_{1}, …, κ_{k−2})κ_{k−1}^{2} + q′′_{k}(κ, κ_{1}, …, κ_{k−2}) ,

with all terms of weight 2k + 2. The main result of this note is:

Proposition 1. In the representation (2.7) of b_{k+1} the coefficient c_{k} 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), as is the number of boundary components. Similarly, b_{1} = c_{1} ∫_{0}^{L} κ^{2} ds so from the constancy of b_{1} on I it follows that I is bounded in L^{2}, as a norm on the curvature. Next consider

(2.9)  b_{2} = c_{2} ∫_{0}^{L} κ_{1}^{2} ds + c′_{2} ∫_{0}^{L} κ^{4} ds ,

using (2.2). The Sobolev embedding theorem shows that for any δ > 0,

∫_{0}^{L} κ^{4} ds ≤ δ ∫_{0}^{L} κ_{1}^{2} ds + c_{δ} ((∫_{0}^{L} κ^{2} ds)^{3} + ∫_{0}^{L} κ^{2} ds) .

Thus from (2.9) and the constancy of b_{2} on I it follows that ∫ κ_{1}^{2} ds is uniformly bounded on I. This in particular implies the boundedness of the supremum norm ∥κ∥_{∞}. Next consider

(2.10)  b_{3} = c_{3} ∫_{0}^{L} κ_{2}^{2} ds + c′_{3} ∫_{0}^{L} κ_{1}^{3} ds + ∫_{0}^{L} q′′_{2}(κ, κ_{1}) ds ,

using (2.2), with q′′_{2} at most quadratic in κ_{1}. 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 b_{0}, b_{1}, b_{2}; thus the uniform boundedness of the L^{2} norm ∥κ_{2}∥_{2} follows from the boundedness of b_{3}.

Proceeding inductively we can suppose that for each p < k, k > 2, the boundedness of b_{0}, b_{1}, … b_{p+1} implies the boundedness of ∥κ_{r}∥_{2}, r ≤ p − 1 and ∥κ_{p}∥_{2}. From (2.7) and (2.8) these estimates imply the boundedness of the second term in (2.7) and hence the boundedness of ∥κ_{k}∥_{2}, and so of ∥κ_{p}∥_{∞} for p ≤ k − 1, given the boundedness of b_{r}, 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′), of the resolvent through

(3.1)  h(t, z, z′) = (2πi)^{−1} ∫_{Γ} e^{−λt}r(λ, z, z′) dλ  z, z′ ∈ Ω, t > 0

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

(3.2)  \begin{cases}(Δ_{z} − λ)r(λ, z, z′) = δ(z, z′) in Ω × Ω \\ r(λ, z, z′) = 0 z ∈ ∂Ω , z′ ∈ Ω . \end{cases}

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′), is not trace class but ∂_{λ}R(λ), with kernel ∂_{λ}r(λ, z, z′) is trace class having continuous kernel. Thus,

(3.3)  H(t) = (2πit)^{−1} ∫_{Γ} ∫ e^{−λt}∂_{λ}r(λ, z, z) dz dλ , t > 0 .

Provided Γ lies outside a conic neighborhood of ℝ̄^{+}, the trace

(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_{ϵ}. The theory of elliptic boundary problems shows that

(3.5)  T^{ϵ}(λ) = ∫_{Ω_{ϵ}} ∂_{λ}r_{ϵ}(λ, z, z) dz

is C^{∞} in ϵ. To compute the coefficient c_{k} in (2.7) it is enough to be able to find the expansion, to all orders, of d^{2}T^{ϵ}(λ)/ dϵ^{2} at ϵ = 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 = ℝ_{x} × ℝ_{y}^{+}. Of course, this is not compact but variations satisfy the same formula, as we shall see. Thus, consider the solution of

(3.6)  \begin{cases}(Δ − λ)r_{ϵ}(λ, z, z′) = δ(z − z′) in Z_{ϵ} × Z_{ϵ} \\ r_{ϵ}(λ, x, ϵτ(x)z′) = 0 \\ r_{ϵ}(λ, z, z′) → 0 as |z| → ∞ . \end{cases}

where Z_{ϵ} = {y > ϵτ(x)} and the boundary perturbation, y = ϵτ(x) has compact support. Now, the arclength differential on the boundary is

(3.7)  ds = (1 + ϵ^{2}(τ′(x))^{2})^{1/2} dx

and the curvature is

(3.8)  x^{ϵ}(x) = ϵτ′′(x)(1 + ϵ^{2}(τ′(x))^{2})^{−1/2} .

Thus, at ϵ = 0, the second variation is

(3.9)  ∂_{ϵ}^{2} ∫_{0}^{∞}(κ_{k}^{ϵ}(s))^{2} ds = 2 ∫_{0}^{∞} (\frac{d^{k+2}}{dx^{k+2}} τ(x))^{2} dx .

If r_{ϵ} is defined by (3.6) then set

(3.10)  r = r_{0}(λ, z, z′) , r = \frac{d}{dϵ}r_{0}(λ, z, z′) , r′′ = \frac{d^{2}}{dϵ^{2}} r_{0}(λ, z, z′)

all kernels on Z × Z, and consider the symbol, outside a cone around ℝ̄^{+} in ℂ

T_{2}(λ) = ∫_{−∞}^{∞} ∫_{0}^{∞} ∂_{λ}r′′(λ, x, y; x, y) dy dx

− ∫_{−∞}^{∞} 2τ(x)∂_{λ}r′(λ, x, 0 ; x, 0) dx (3.11)

+ ∫_{−∞}^{∞} τ^{2}(x)(∂_{y} + ∂_{y′})∂_{y}r(λ, x, 0 ; x, 0) dx .

Proposition 2. T_{2}(λ) has a classical expansion as |λ| → ∞ outside any cone containing ℝ̄^{+}, of the form

(3.12)  T_{2}(λ) ∼ ∑_{p≥1} e_{p}(λ) ∫_{−∞}^{∞} (\frac{d^{p}τ}{dx^{p}} )^{2} dx + f(λ)τ(x) dx

where the order of e_{p} tends to −∞ as p → ∞. The coefficients e_{p}(λ) are, modulo S^{−∞}, the same as would be obtained by expansion of d^{2}T^{ϵ}(λ)/dϵ^{2}, at ϵ = 0, for the same perturbation y = ϵτ(x) of the boundary y = 0 of the cylindrical domain ℝ_{x}/pℤ × [0, 1] for p so large that supp(τ) ⊂ (−\frac{1}{2}p , \frac{1}{2}p).

Proof. Consider the compact manifold, with flat metric,

X = ℝ_{x}/pℤ × [0, 1]

under perturbation to X_{ϵ}, with the boundary y = 0 modified to y = ϵτ(x), −\frac{1}{2}p ≤ x ≤ \frac{1}{2}p being considered as a fundamental domain for ℝ/pℤ. The existence of an expansion for T^{ϵ}(λ), given by (3.5) is standard, with the coefficients again being the invariants a_{k} , b_{k} of (1.1). The existence of an expansion (3.12) for d^{2}T^{0}(λ)/dϵ^{2} therefore follows from formulae (3.7), (3.8) but now for perturbation X_{ϵ} of X. Only terms linear or quadratic in τ can occur, so the expansion must be of the form (3.12). Moreover, if r̄^{ϵ}(λ, z, z′) is the corresponding kernel, satisfying

(3.13)  \begin{cases}(Δ − λ)r̄^{ϵ}(λ, z; z′) = δ(z − z′)X_{ϵ} × X_{ϵ} \\ r̄^{ϵ}(λ, z, z′) = 0 y = ϵτ(x) on y = 1 \end{cases}

then from (3.5),

(3.14)  T (λ)=∫_{− \frac{1}{2} p}^{\frac{1}{2} p} ∫_{ϵτ(x)}^{1} ∂_{λ}r̄^{ϵ}(λ, x, y; x, y) dy dx .

Direct differentiation gives:

d^{2}T^{0}/dϵ^{2} = ∫_{X} ∂_{λ}r̄′′(λ, z, z) dz − 2 ∫_{− \frac{1}{2} p}^{\frac{1}{2} p} τ(x)∂_{λ}r̄′(λ, x, 0 ; x, 0) dx (3.15)

− ∫_{− \frac{1}{2} p}^{\frac{1}{2} p} τ^{2}(x)(∂_{y} + ∂_{y′})∂_{λ}r̄(λ, x, 0 ; x, 0) dx

where r̄, r̄′, r̄′′ are defined as in (3.10) but from r̄^{ϵ}.

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^{ϵ} satisfying (3.6) to the corresponding kernel r̃^{ϵ} where ℝ_{x} is replaced by ℝ_{x}/pℤ, assuming always that the perturbation has support in (−\frac{1}{2}p , \frac{1}{2}p). In fact,

r̃^{ϵ}(λ, z, z′) = ∑_{k∈ℤ}r^{ϵ}(λ, z + kp, z′) .

Standard elliptic estimates show that on X_{ϵ} the difference

(3.16)  ∂_{ϵ}r̃^{ϵ} − ∂_{ϵ}r^{ϵ} = 0 (|λ|^{−N}) ∀ N .

Similarly, in passing from r̃^{ϵ} to r̄^{ϵ} 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 ϵ, 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 d^{2}T^{0}/ dϵ^{2} , the coefficients can be related to the c_{k} 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, r′, r′′ and associated kernels. If (ξ^{2} − λ)^{1/2} is for ξ ∈ ℝ, λ ∈ ℝ̄^{+} the branch with positive real part, then (4.1)

F (λ, z, z′) = \begin{cases}(4π)^{−1} ∫ e^{i(x−x′)ξ−(ξ^{2}−λ)^{1/2}(y−y′)}(ϵ^{2} − λ)^{−1/2} dξ , y > y′ \\ (4π)^{−1} ∫ e^{i(x−x′)ξ−(ξ^{2}−λ)^{1/2}(y′−y)}(ϵ^{2} − λ)^{−1/2} dξ , y′ > y \end{cases}

is the preferred, i.e., temperate, fundamental solution of Δ − λ on ℝ^{2}. In terms of this kernel,

(4.2) r(λ, z, z′) = F (λ, z, z′) − G(λ, z, z′)
(4.3) G(λ, z, z′) = (4π)^{−1} ∫ e^{i(x−x′)ξ−(ϵ^{2}−λ)^{1/2}(y+y′)}(ξ^{2} − λ)^{−1/2} dξ .

Thus, by continuity

(4.4)  (∂_{y} + ∂_{y′})∂_{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

(4.5)  \begin{cases}(Δ − λ)P (λ, z; x′) = 0 \\ P (λ, x, 0 ; x′) = δ(x − x′) \end{cases}

and vanishing at infinity is

(4.6)  P (λ, z; x′) = (2π)^{−1} ∫ e^{i(x−x′)ξ−(ξ^{2}−λ)^{1/2}y} dξ .

Differentiating (3.6) gives the boundary problem for r′:

(4.7) (Δ − λ)r′(λ, z; z′) = 0 , r′(λ, x, 0 ; z′) = −τ(x)∂_{y}r(λ, x, 0 ; z′) ,

with r′ vanishing at infinity. Thus, from (4.6) and (4.2),

r′(λ, z; z′) = lim_{χ→1} −(2π)^{−2} ∫ e^{i(x−x′)ξ+i(x′′−x)ξ′−(ξ^{2}−λ)^{1/2}y−((ξ′)^{2})^{1/2}y′} (4.8)

·τ(x′′)χ(ξ)χ(ξ′) dξ′ dξ dx′′ .

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 r′′:

(4.9) \begin{cases}(Δ − λ)r′′(λ, z; z′) = 0 \\ r′′(λ, x, 0 ; z′) = −2τ(x)∂_{y}r′(λ, x, 0 ; z′) − τ^{2}(x)∂_{y}^{2}r(λ, x, 0 ; z′) . \end{cases}

From (4.2), ∂_{y}^{2}r = 0, at y = 0, z′ ∈ Z so the second term in the boundary condition for r′′ is zero. Applying (4.6) again allows one to construct r′′, we only need

∂_{λ}r′′(λ, z, z) = lim_{χ→1} ∂_{λ} − 2(2π)^{−3} ∫ e^{i(x−x′)ξ+i(x′−x′′)ξ′+i(x′′−x)ξ′′} (4.10)

·e^{−(ξ^{2}−λ)^{1/2}y−((ξ′′)^{2}−λ)^{1/2}y}((ξ′)^{2} − λ)^{1/2}
·τ(x′)τ(x′′)χ(ξ)χ(ξ′)χ(ξ′′) dξ′′ dξ′ dξ dx′′ dx′ .

Again from (4.8), the second term in (3.11) is identically zero, so

T_{2}(λ) = lim_{χ→1} ∂_{λ} − 2(2π)^{−3} ∫ e^{i(x−x′)ξ+i(x′−x′′)ξ′+i(x′′−x)ξ′′} (4.11)

·τ(x′)τ(x′′)χ(ξ)χ(ξ′)χ(ξ′′)((ξ′)^{2} − λ)^{1/2}((ξ^{2} − λ)^{1/2} + ((ξ′′)^{2} − λ)^{1/2})^{−1}

· dξ dξ′ dξ′′ dx′ dx′′ dx .

In (4.11) the integral over x, ξ′′ can be evaluated directly, since the integrand is independent of x, giving:

(4.12) T_{2}(λ) = lim_{λ→ξ} ∂_{λ} − (2π)^{−2} ∫ e^{i(x′−x)(ξ′−ξ)}τ(x′)τ(x)
·χ(ξ)χ(ξ′)((ξ′)^{2} − λ)^{1/2}(ξ^{2} − λ)^{−1/2} dξ dξ′ dx′ dx .

The integrals over x′ and ξ′ can now be evaluated by the lemma of stationary phase:

(4.13) T_{2}(λ) ∼ ∑_{k≥1} e_{k}(λ) ∫ (d^{k}τ(x)/ dx^{k})^{2} dx

e_{k}(λ) = −((2k)!(2π))^{−1}∂_{λ} ∫ (ξ^{2} − λ)^{−1/2}∂_{ξ}^{2k}(ξ^{2} − λ)^{1/2} dξ

the odd order terms being automatically zero.

The coefficient e_{k}(λ) is actually homogeneous of degree −k − \frac{1}{2}, so from its holomorphy properties

(4.14)  e_{k}(λ) = γ_{k−2}(−λ)^{−k− \frac{1}{2}} ,

with

γ_{k−2} (4.15)

= − (k − \frac{1}{2}) ((2k)!(2π))^{−1} ∫ (s^{2} + 1)^{−1/2}d^{2k}(s^{2} + 1)^{1/2}/ ds^{2k} ds

= (−1)^{k} (k − \frac{1}{2})^{2} ((2k)!2π)^{−1} ∫ (d^{k−1}(s^{2} + 1)^{−1/2}/ ds^{k−1})^{2} ds .

This shows that all the coefficients γ_{k−2}, k ≥ 2, are non-zero. Tracing backwards through (3.12) it follows easily that all the c_{k}, k ≥ 0, are nonzero, providing Proposition 1.

More explicitly a short computation gives the formula

(4.16)  c_{k} = \frac{1}{2}γ_{k} \frac{1}{2π} ∫_{Γ} e^{−λ_{(−λ)}−k−5/2} dλ  k > 0 ,

using (4.15) this becomes: (4.17)

c_{k} = Γ(−k − 3/2)(k + 3/2)^{2}(4π^{2} · (2k + 4)!)^{n} ∫ (d^{k+1}(1 + s^{2})^{−1/2}/ ds^{k+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.