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Transplantation of Eigenfunctions on Isospectral Domains
Feynman Liang
Dartmouth College
Background
In 1966, Mark Kac published an article in the American Mathematical Monthly asking “Can
one hear the shape of a drum?” [1]. Kac’s question is not actually about a three-dimensional
drum, but instead a two dimensional drumhead modeled as a planar bounded domain with
smooth boundary Γ. Wave propagation on Γ is described by the wave equation:
∂2f
∂t2
= ∆f (1)
where z = f(x, y, t) represents the drumhead’s displacement at time t of the point (x, y) ∈ Γ.
Since the drumhead is attached at the boundaries, we impose homogeneous Dirichlet bound-
ary conditions:
f(x, y, t) = 0 for all t ∈ R and (x, y) ∈ ∂Γ (2)
Separation of variables is a method for solving this PDE. Assume f(x, y, t) = g(x, y)h(t) where
g(x, y) represents a standing wave and h(t) the time evolution of the wave. Substituting into
Equation 1, we have:
g(x, y)h (t) = (∆g(x, y))h(t)
Rewrite this as
h (t)
h(t)
=
∆g(x,y)
g(x,y)
and notice that since the left-hand side is independent of (x, y)
and the right-hand side is independent of t, both must be equal to some separation constant
−λ ∈ R. This yields Laplace’s Equation:
∆g = −λg (3)
h (t) = −λh(t) (4)
A function g : Γ → R which satisfies Equation 2 and Equation 3 is called a Dirichlet eigen-
function of the Laplacian on Γ with eigenvalue −λ. The Dirichlet spectrum of Γ is the set
of all λ for which there exists a Dirichlet eigenfunction. Notice that a solution to Equation 4
h(t) = Kei
√
λt is harmonic in time with frequency
√
λ, representing a pure tone the drum-
head is capable of producing (via the normal mode g(x, y)). We interpret hearing a drum as
knowing the Dirichlet spectrum of Γ, so Kac’s question is really asking if spectrum determines
shape.
In 1992, Gordon, Webb, and Wolpert constructed a pair of drumheads with different shapes
(nonisometric) but identical spectrums (isospectral) [2], providing a planar counterexample to
Kac’s question. In this poster, we will explore a method called transplantation for proving two
planar domains have the same spectrum and verify the results with a discretized numerical
computation.
Transplantation of Waveforms
To show two planar domains have the same spectrum, we will show how to transplant an
eigenfunction of one domain to an eigenfunction with identical eigenvalue on the other do-
main. Here we consider only smooth eigenfunctions, choosing to neglect weak solutions.
Figure 1 shows a standard example of two isospectral domains[3]. Suppose ϕ is a Dirichlet
eigenfunction on D1. Divide D1 and D2 into an equal number of congruent triangles, and
consider the restrictions of ϕ to each of the triangles. Note that the restriction of ϕ still sat-
isfies ∆ϕ = λϕ, except now with possibly nonzero boundary values. However, smoothness
of ϕ requires coinciding triangle edges to agree smoothly and the zero Dirichlet boundary
condition on both domains requires that triangle edges on ∂D1 and ∂D2 must be zero.
(a) D1 (b) D2
Figure 1: Example of two isospectral domains
By linearity of Equation 3, a linear combination of solutions ϕ is still a solution. Hence, we
can piece together the restrictions of ϕ onto the triangles of D1 to transplant ϕ into a solution
ψ on D2. To ensure ψ is smooth, we must be careful to ensure the pieced together solution
satisfy:
S.1 On edges between two adjacent triangles, ψ smooth.
S.2 On edges contained in ∂D2, ψ = 0.
These conditions can be easily verified on Figure 1. We will illustrate using the triangle
−B + E − F in D2. Continuity ensures that ϕ takes the same boundary value on the shared
blue edge of F and E. Hence, the sum of ϕ on E with the −ϕ on F is zero on the blue edge
and by linearity still has eigenvalue λ. As the blue edge of B is on ∂D1, the ϕ to B is also zero
on this boundary. Thus, the combination −B + E − F is identically zero on the blue boundary
which is also an edge on ∂D2, satisfying S.2.
S.1 can be similarly checked; we will illustrate the red edge between −B + E − F and
−A + D + F is compatible. As the red edge of F is on the boundary of D1, ϕ is zero on this
boundary. Thus, to check compatibility in D2 it suffices to show that −A + D and −B + E
share the same value on the red edge boundary. Notice that in D1 the triangles D and E
share a red edge as do the triangles A and B, hence by S.1 on D1 ϕ takes on the same value
for points on the red edges of D and E (and analogously for A and B). Therefore, −A + D’s
red edge has the same value as −B + E and ψ is continuous. Smoothness can be shown
using the reflection property of Dirichlet eigenfunctions to fully verify S.1 on D2.
Proceeding in this manner, transplantation is able to take an eigenfunction ϕ on D1 and
return an eigenfunction ψ on D2 with the same eigenvalue and satisfying boundary condi-
tions. The converse (obtaining ϕ from ψ) can also be done by solving a linear system of 7
equations with 7 unknowns. Thus, the eigenvalues of D1 and D2 are identical so the two
domains are isospectral.
Numerical Verification
To check that we have successfully constructed two isospectral domains, we will approximate
the Laplacian using finite difference and compute the spectrum of the discretized problem.
We first discretize the two domains, yielding:
Figure 2: Discretized drumhead domains
A function g on this discretized domain is given by its value at each point, which can be viewed
as a finite dimensional vector g. In this discrete setting, the Laplacian ∆g(x, y) takes the form:
∆g(x, y) =
g(x + h, y) − 2g(x, y) + g(x − h, y)
h2
+
g(x, y + h) − 2g(x, y) + g(x, y − h)
h2
(5)
where h is the distance between adjacent grid points. Equation 5 is a linear combination of
points {g(x, y), g(x±h, y), g(x, y±h)} and can be written as a matrix A operating on g, allowing
us to rewrite Equation 3:
∆g = λg ←→ A g = λg
Computing the eigenvectors of A will then yield the value of eigenfunctions at the grid points.
If the two domains are truly isospectral, then the set of eigenvalues of A should be the same
for both domains. Carrying out this computation, Figure 3 shows the first eight (truncated)
eigenvalues[4].
D1 D2
10.1659 10.1659
14.6306 14.6306
20.7176 20.7176
26.1151 26.1151
28.9835 28.9835
36.7741 36.7741
42.2830 42.2830
46.0342 46.0342
Figure 3: First eight eigenvalues of the discretized Laplacian on D1 and D2
Notice that the eigenvalues agree up to numerical roundoff, consistent with the fact that the
two domains are isospectral. In Figure 4 we visualize the corresponding eigenfunctions by
linearly interpolating between grid points.
Figure 4: Eigenfunctions of the discretized Laplacian on D1 and D2, linearly interpolated
Of particular interest is the eigenfunction corresponding to the 9th eigenvalue, which consists
of solutions on the basic isosceles triangle reflected to cover the entire domain. Transplanta-
tion from D1 to D2 can be easily verified here by comparing the polarity (indicated by color in
Figure 4) of each triangle patch.
References
[1] Marc Kac, “Can one hear the shape of a drum?”, Amer. Math. Monthly 73 (1966), 1-23.
[2] Carolyn Gordon, David Webb, Scott Wolpert, “One cannot hear the shape of a drum”, Bull.
Amer. Math. Soc. 27 (1992), 134-138.
[3] Gordon, Carolyn; Webb, David, “You can’t hear the shape of a drum”, American Scientist
84 (January–February): 4655
[4] Moler, Cleve. “Can One Hear the Shape of a Drum?” Cleves Corner: Cleve Moler on
Mathematics and Computing. The MathWorks, Inc., 6 Aug. 2012. Web. 05 Mar. 2015.

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