Spectral Determination of Convex Planar Domains

Spectral Determination of Convex Planar Domains

Definitions and mathematical statement

For a bounded open convex domain $\Omega\subset{\mathbb{R}}^2$, the Dirichlet Laplacian is the self-adjoint operator associated with the quadratic form $u\mapsto\int_\Omega|\nabla u|^2$ on $H_0^1(\Omega)$. Write its discrete spectrum, with multiplicities, as \[ 0<\lambda_1(\Omega)\le\lambda_2(\Omega)\le\cdots. \] The question is whether \[ \lambda_j(\Omega_1)=\lambda_j(\Omega_2)\ (j\ge1) \quad\Longrightarrow\quad \Omega_2=U\Omega_1+b \] for an orthogonal matrix $U$ and translation $b$. Reflections are included among congruences. The entire spectrum with multiplicities is required; equality of finitely many eigenvalues or heat invariants is weaker. The convexity assumption must be retained when considering examples of isospectral domains.

Formulation and concept references: [S1].

Short English statement

Does the full Dirichlet spectrum determine a bounded convex planar domain uniquely up to Euclidean congruence?

Research attempt

Scope and source audit.

This exploratory attempt by GPT-6 Astra Ultra was prepared on 27 September 2026. The target is global uniqueness among all bounded open convex planar domains for the full Dirichlet spectrum with multiplicities. A statement restricted to a family, to an infinitesimal deformation, or to finitely many spectral invariants is a smaller assertion. The supplied survey [S1], Section 7, identifies the convex-drum question as open. The source check also found the 2026 manuscript [E2] claiming strictly convex isospectral examples for the Steklov problem. Those concern a different operator and are not counterexamples to the present statement. No argument from that manuscript is used below.

The relevant heat-trace input is the two-term Dirichlet asymptotic on bounded Lipschitz domains, discussed with its precise regularity in [E1]. Bounded convex planar domains with nonempty interior have Lipschitz boundary, so that input applies even when their boundaries are not smooth. Higher smooth-boundary coefficients are used only for the explicit smooth examples below. This distinction prevents a silent smoothness assumption from narrowing the original question.

Attack map.

First extract geometric data from the spectrum and solve a restricted reconstruction problem exactly. Then try to reconstruct an arbitrary convex body from the area and perimeter. An explicit pair of noncongruent strictly convex analytic boundaries will disprove that intermediate inference. Finally retain the complete heat trace instead of its initial coefficients, and locate the genuinely missing inverse step. These are separate proof and counterexample tracks: a counterexample to a proposed finite-invariant criterion is not an isospectral counterexample.

Lemma 492.1: the first two heat coefficients are audible.

Define \[ Z_\Omega(t)=\sum_{j=1}^{\infty}e^{-t\lambda_j(\Omega)}, \qquad t>0. \] For a bounded Lipschitz planar domain the established heat expansion is \begin{equation}\label{wsref-0-61-34-39-32-3a-68-65-61-74} Z_\Omega(t)=\frac{A}{4\pi t} -\frac{P}{8\sqrt{\pi t}}+o(t^{-1/2}),\qquad t\downarrow0, \end{equation} where $A=|\Omega|$ and $P=\mathcal H^1(\partial\Omega)$. This is the two-term heat formula, not a pointwise two-term counting law requiring additional dynamical hypotheses. See [E1], equation $3$ and its discussion of Brown's Lipschitz-boundary theorem. We use that analytic theorem as an external input, not as a new result proved here.

The extraction itself is immediate but worth making exact: \begin{equation}\label{wsref-0-61-34-39-32-3a-65-78-74-72-61-63-74} A=\lim_{t\downarrow0}4\pi t Z_\Omega(t),\qquad P=-8\sqrt\pi\lim_{t\downarrow0}\sqrt t \left(Z_\Omega(t)-\frac{A}{4\pi t}\right). \end{equation} Thus isospectral convex domains have equal area and perimeter. For smooth simply connected planar domains the next constant coefficient is $1/6$; the curvature form is $(12\pi)^{-1}\int_{\partial\Omega}\kappa\,ds$, and the total turning is $2\pi$. The smooth heat formula and the distinction from corner coefficients are discussed in [S1], Section 4.2.

In the equality case $P^2=4\pi A$, the planar isoperimetric theorem [E3] forces the domain to be a disk. Hence a disk is determined among the convex domains in the question by these two audible quantities alone. This uses the classical equality case of the isoperimetric theorem as an external geometric input. Away from equality, that argument supplies no uniqueness.

Attempt A: exact reconstruction inside the rectangle family.

Let $\Omega=(0,a)\times(0,b)$ with $a\ge b>0$. Separation of variables gives the complete Dirichlet eigenvalue multiset \begin{equation}\label{wsref-0-61-34-39-32-3a-72-65-63-74-61-6e-67-6c-65} \lambda_{m,n}=\pi^2\left(\frac{m^2}{a^2}+\frac{n^2}{b^2}\right), \qquad m,n\ge1. \end{equation} The products of one-dimensional sine bases are a complete orthogonal basis of $L^2(\Omega)$, so there are no missing eigenfunctions. Put $\alpha=a^{-2}$ and $\beta=b^{-2}$; then $0<\alpha\le\beta$. The smallest index pair is $(1,1)$. Among every other pair, either $m\ge2$, giving a value at least $4\alpha+\beta$, or $n\ge2$, giving a value at least $\alpha+4\beta\ge4\alpha+\beta$. Consequently, counting multiplicity, \[ \lambda_1=\pi^2(\alpha+\beta),\qquad \lambda_2=\pi^2(4\alpha+\beta). \] Solving the two linear equations proves \begin{equation}\label{wsref-0-61-34-39-32-3a-69-6e-76-65-72-73-65} a=\pi\sqrt{\frac{3}{\lambda_2-\lambda_1}},\qquad b=\pi\sqrt{\frac{3}{4\lambda_1-\lambda_2}}. \end{equation} For a square the second eigenvalue is double, but its value is still the one used above. The denominators are positive. The ratio satisfies $1<\lambda_2/\lambda_1\le5/2$, with equality at $5/2$ exactly for $a=b$. This proves uniqueness within the rectangle family from just two eigenvalues. It does not show that an arbitrary convex isospectral competitor must be a rectangle.

There is also a reconstruction from the two heat coefficients: $a+b=P/2$ and $ab=A$, so $a,b$ are the roots of $z^2-(P/2)z+A=0$. Its discriminant is $(a-b)^2$. This reveals a stability limitation near a square. Small errors in $A,P$ can produce square-root errors in the recovered side difference. Exact uniqueness does not by itself provide a uniformly well-conditioned numerical inverse, and a finite precision eigenvalue match is not an exact spectral identity.

Attempt B: test whether area and perimeter determine a convex body.

We now give an explicit obstruction to extending the preceding two-invariant argument. For an integer $m\ge2$, let \begin{equation}\label{wsref-0-61-34-39-32-3a-73-75-70-70-6f-72-74} h_m(\theta)=R+a_m\cos(m\theta),\qquad a_m=\frac{\tau}{\sqrt{m^2-1}}, \quad 0<\tau<\frac{R}{\sqrt{m^2-1}}. \end{equation} With $u=(\cos\theta,\sin\theta)$ and $u_\perp=(-\sin\theta,\cos\theta)$, define \[ r_m(\theta)=h_m(\theta)u(\theta)+h'_m(\theta)u_\perp(\theta). \] Its derivative is \[ r'_m(\theta)=\rho_m(\theta)u_\perp(\theta),\qquad \rho_m=h_m+h''_m=R-\tau\sqrt{m^2-1}\cos(m\theta)>0. \] This is a regular analytic strictly convex closed curve with support function $h_m$. Here is a direct check of the support assertion. For fixed $\phi$, the derivative of $r_m(\theta)\cdot u(\phi)$ equals $\rho_m(\theta)\sin(\phi-\theta)$, which is negative on $(\phi,\phi+\pi)$ and positive on $(\phi+\pi,\phi+2\pi)$. It therefore has a unique maximum at $\theta=\phi$ modulo $2\pi$, with value $h_m(\phi)$. Each direction has its unique supporting point; the curve traces the boundary of its convex hull once and is embedded. The strictly positive radius of curvature excludes flat segments.

Its perimeter and area follow without a numerical approximation: \begin{align} P_m&=\int_0^{2\pi}\rho_m(\theta)\,d\theta=2\pi R, \label{wsref-0-61-34-39-32-3a-70-65-72-69-6d-65-74-65-72}\\ A_m&=\frac12\int_0^{2\pi}\det(r_m,r'_m)\,d\theta =\frac12\int_0^{2\pi}(h_m^2-(h'_m)^2)\,d\theta =\pi R^2-\frac\pi2\tau^2. \label{wsref-0-61-34-39-32-3a-61-72-65-61} \end{align} The integration by parts uses periodicity, and the final equality uses the zero average of the cosine and the integrals $\int\cos^2(m\theta) =\int\sin^2(m\theta)=\pi$.

Choose $R=1$, $\tau=1/10$, and $m=2,3$. Both curvature radii are positive, and the two interiors have identical area $199\pi/200$, perimeter $2\pi$, and the same constant smooth heat coefficient $1/6$. They are not congruent. The $m=2$ body is centrally symmetric since $h_2(\theta+\pi)=h_2(\theta)$. If the $m=3$ body had centre $c$, its translated support function would be $\pi$-periodic, requiring \[ 2a_3\cos(3\theta)=2c\cdot u(\theta). \] Multiply by $\cos(3\theta)$ and integrate over a period. The right side is zero by Fourier orthogonality while the left side is $2\pi a_3>0$. This contradiction excludes every possible centre. Since central symmetry is preserved by translations, rotations, and reflections, the bodies cannot be congruent.

This pair refutes reconstruction from the first three displayed heat coefficients even in the analytic, strictly convex class. It is expressly not asserted to be isospectral. A further geometric diagnostic already distinguishes them: \begin{equation}\label{wsref-0-61-34-39-32-3a-63-75-72-76-61-74-75-72-65} \int_{\partial\Omega_m}\kappa^2\,ds =\int_0^{2\pi}\frac{d\theta}{\rho_m(\theta)} =\frac{2\pi}{\sqrt{R^2-(m^2-1)\tau^2}}. \end{equation} The last integral follows by periodic substitution and the tangent half-angle substitution in $\int_0^{2\pi}(R-c\cos\theta)^{-1}d\theta$. This calculation shows that matching the initial data has not even fixed all elementary curvature statistics. No coefficient of a higher heat term is needed for the counterexample to the proposed two-invariant method.

Attempt C: retain the full heat trace.

An equality $Z_{\Omega_1}(t)=Z_{\Omega_2}(t)$ for every $t>0$ is equivalent to equality of the complete eigenvalue multisets. To verify the nontrivial direction, write the lowest distinct eigenvalue as $\mu_1$ with multiplicity $k_1$. Then \[ \mu_1=-\lim_{t\to\infty}t^{-1}\log Z(t),\qquad k_1=\lim_{t\to\infty}e^{t\mu_1}Z(t). \] The remaining tail tends to zero after multiplication by $e^{t\mu_1}$: for fixed $t_0>0$, dominate its terms for $t\ge t_0$ by the summable series $e^{-t_0(\lambda_j-\mu_1)}$ and use dominated convergence. Subtract $k_1e^{-t\mu_1}$ and repeat to recover each successive distinct eigenvalue and its multiplicity. Discreteness and divergence of the Dirichlet eigenvalues justify this induction. Thus there is no loss of information in switching from the whole spectrum to the whole heat trace.

There is a loss in replacing that function by a finite, or merely formal, small-time expansion. Even two abstract functions whose difference is $e^{-1/t}$ have identical power asymptotics to every order at zero while being unequal for $t>0$. This elementary example is not claimed to be a pair of heat traces of domains. It pinpoints why coefficient matching alone needs an additional uniqueness theorem before it can substitute for the full spectral data.

In support coordinates the actual missing implication is now concrete: for arbitrary convex bodies with support functions $h_1,h_2$, prove that equality of their Dirichlet heat traces for all positive times forces \[ h_2(\theta)=h_1(\varepsilon\theta-\theta_0)+b\cdot u(\theta), \qquad \varepsilon\in\{1,-1\}. \] This is precisely the action of orthogonal transformations and translations on support functions. The formulas (a492:perimeter)--(a492:area) access only two averages of $h$; no argument here extracts its full angular dependence from $Z(t)$. Positivity of $h+h''$ restricts the class but does not repair the information loss, as the explicit pair shows.

Verification, boundary cases, and outcome.

The companion script uses rational arithmetic to enumerate rectangle eigenvalues $\lambda_{m,n}/\pi^2$ for integer sides from 1 through 12. For every unordered pair of sides it checks the first two sorted eigenvalues and the exact inverse formulas, including the square's multiplicity. It also checks the algebraic curvature-radius positivity for the two support functions and records numerical quadrature as a diagnostic of the derived area, perimeter, and curvature integrals. Quadrature is not used to prove convexity, noncongruence, or equality of any spectrum.

The zero-width rectangle is excluded since the target is an open planar domain. Reflections are included throughout. The support-function examples are interiors of compact convex bodies with positive curvature and hence satisfy all domain hypotheses. The global problem also allows corners, so an eventual general proof cannot rely solely on their analytic parametrizations. Equality of a finite number of eigenvalues, or of values computed to finite precision, would not satisfy its hypothesis.

UNRESOLVED. We have an exact restricted reconstruction theorem, an attributed disk rigidity argument, a completely explicit failure of reconstruction from the first heat invariants, and an exact equivalence between complete heat data and complete spectral data. These do not prove global uniqueness for convex planar domains. The next substantive tasks are to extract angular boundary information from spectral quantities beyond the initial heat coefficients, to prove a rigidity theorem with hypotheses covering all convex competitors, or to construct two noncongruent convex domains with an exact full-spectrum identity. The examples here accomplish none of these final steps and are not proposed counterexamples to the catalogue conjecture.

Sources

• S1 G. Mardby and J. Rowlett, 112 years of listening to Riemannian manifolds, section on planar domains $2024$. https://arxiv.org/html/2406.18369v2. Formulation source.
• E1 Rupert L. Frank and Simon Larson, On the error in the two-term Weyl formula for the Dirichlet Laplacian, arXiv:2001.01876, Introduction, equation $3$ and discussion of Brown's two-term heat formula for Lipschitz domains. https://arxiv.org/html/2001.01876. External heat-asymptotic input; consulted 27 September 2026.
• E2 Tao Hu, Jiachen Shi, and Quanyu Tang, Strictly Convex Steklov-Isospectral Plane Domains, arXiv:2608.10557v2, 15 August 2026, abstract. https://arxiv.org/abs/2608.10557v2. Operator-scope check only: this manuscript concerns Steklov, not Dirichlet, spectra. Its construction is not used here.
• E3 Robert Osserman, The isoperimetric inequality, Bulletin of the American Mathematical Society 84 $1978$, 1182--1238, opening statement of the planar inequality and equality case. https://sites.math.washington.edu/~toro/Courses/20-21/MSF/osserman.pdf. Classical geometric input for the disk subcase; consulted 27 September 2026.