[#P2922] Planar drums whose spectra differ only finitely

A mathematical schematic of Planar drums whose spectra differ only finitely. — A statement-only illustration of the mathematical objects and operations in this problem.

Problem. Do there exist two bounded connected planar domains \(\Omega_1,\Omega_2\) with \(C^\infty\) boundaries and an index \(N\) such that their Dirichlet eigenvalues, listed nondecreasingly with multiplicity, satisfy \(\lambda_k(\Omega_1)=\lambda_k(\Omega_2)\) for every \(k\ge N\), while the full spectra differ?

1Context

Heat-trace identities, wave invariants, and transplantation data are reusable across candidate pairs. The finite discrepancy produces an explicit finite exponential sum in the difference of heat traces, which offers a rigid analytic target.

2Problem setup

Definition 1. The Dirichlet eigenvalues are the eigenvalues of -Delta on a domain with zero boundary values, repeated according to multiplicity.

Remark 1. The full spectra differ when lambda_j(Omega_1) differs from lambda_j(Omega_2) for at least one index j<N.

3What counts as a solution

Construct two domains and prove exact equality of every eigenvalue from some index onward while certifying at least one earlier mismatch, or prove that eventual equality forces equality of the complete Dirichlet spectra.
A construction must specify the domains sufficiently to prove smoothness, connectedness, and exact spectral claims; numerical eigenvalue agreement is insufficient.

1Status

Current status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: The MathOverflow thread gives no construction or rigidity proof for smooth planar domains. Nearby work treats full isospectrality and eventual spectral agreement in other geometric categories. Exact unresolved remainder: Construct two domains and prove exact equality of every eigenvalue from some index onward while certifying at least one earlier mismatch, or prove that eventual equality forces equality of the complete Dirichlet spectra. A construction must specify the domains sufficiently to prove smoothness, connectedness, and exact spectral claims; numerical eigenvalue agreement is insufficient.^[3][2][1]

1Records

Record	Kind	Assessment
	Result	Supported
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: The MathOverflow thread gives no construction or rigidity proof for smooth planar domains. Nearby work treats full isospectrality and eventual spectral agreement in other geometric categories. Exact unresolved remainder: Construct two domains and prove exact equality of every eigenvalue from some index onward while certifying at least one earlier mismatch, or prove that eventual equality forces equality of the complete Dirichlet spectra. A construction must specify the domains sufficiently to prove smoothness, connectedness, and exact spectral claims; numerical eigenvalue agreement is insufficient.^[3][2][1] Relevance to this problem For Planar drums whose spectra differ only finitely, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 3, Reference 2, Reference 1 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: The MathOverflow thread gives no construction or rigidity proof for smooth planar domains. Nearby work treats full isospectrality and eventual spectral agreement in other geometric categories. Exact unresolved remainder: Construct two domains and prove exact equality of every eigenvalue from some index onward while certifying at least one earlier mismatch, or prove that eventual equality forces equality of the complete Dirichlet spectra. A construction must specify the domains sufficiently to prove smoothness, connectedness, and exact spectral claims; numerical eigenvalue agreement is insufficient. Open full record →
	Trace	Reported
attempt · Route 1 Construct two domains and prove exact equality of every eigenvalue from some index onward while certifying at least one earlier mismatch, or prove that eventual equality forces equality of the complete Dirichlet spectra. A construction must specify the domains sufficiently to prove smoothness, connectedness, and exact spectral claims; numerical eigenvalue agreement is insufficient.^[1] Relevance to this problem Gives the next worker a direct completion target while separating partial progress from a full answer. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Work against the displayed statement and preserve every hypothesis and quantifier. Construct two domains and prove exact equality of every eigenvalue from some index onward while certifying at least one earlier mismatch, or prove that eventual equality forces equality of the complete Dirichlet spectra. A construction must specify the domains sufficiently to prove smoothness, connectedness, and exact spectral claims; numerical eigenvalue agreement is insufficient. Any computation must retain a replayable witness and a matching exclusion or completeness certificate. Open full record →
No records match these filters.

Notes and companion materialContext, examples, and computations

Original intake status. UNKNOWN as of 2026-07-27. The MathOverflow thread gives no construction or rigidity proof for smooth planar domains. Nearby work treats full isospectrality and eventual spectral agreement in other geometric categories.

Planar isospectral constructions, including arXiv:1005.1839, were checked for examples with altered low eigenvalues. Their constructed pairs have identical full spectra and therefore do not answer the question.
A 2026 preprint on eventual spectral agreement for locally symmetric manifolds was checked as nearby rigidity evidence. Its hypotheses do not cover bounded Euclidean planar domains with boundary.
A local corpus search for eventual isospectrality, finite spectral difference, and planar drums found no duplicate.
Independent source, duplicate, exact-title, and equivalent-formulation review completed on 2026-08-01.

Recorded example 1. A standard isospectral pair satisfies eventual equality with N=1 and therefore fails the required full-spectrum mismatch.

2See also

Hot spots conjecture for convex planar domainsspectral geometry
Simple Laplace spectrum for a generic metric in a Kahler classspectral geometry
Three-dimensional Euler regularitypartial differential equations

How to cite

TheoremDB contributors, “Planar drums whose spectra differ only finitely,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/planar-drums-finitely-different-spectra

This page as plain text: planar-drums-finitely-different-spectra.md

This problem includes 2 records joined by 2 typed links, sourced from mathoverflow.net^[1], current as of August 1, 2026.

1References

Packet source. MathOverflow question 474950, “Planar drums whose spectra differ only finitely,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗forum · reference source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at Question 474950 and every visible answer and comment were checked on 2026-07-27.Also cited at Full question, answers, and visible comments concerning Planar drums whose spectra differ only finitely; checked 2026-08-01.Also cited at Editorial research route recorded 2026-08-01.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Planar drums whose spectra differ only finitely, the reviewed source scope is Full question, answers, and visible comments concerning Planar drums whose spectra differ only finitely; checked 2026-08-01.. The packet makes no inference beyond that cited scope.Source named by the research packet.
Sudhir Pujahari and Punya Plaban Satpathy, “Near Isospectrality and Spectral Rigidity for Compact Locally Symmetric Manifolds,” arXiv:2606.09320 (2026). Question statement, visible answers and comments, or the linked article sections described in the source record. ↗preprint · primary source · arXiv:2606.09320, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at abstract and spectral-rigidity theorem for compact locally symmetric manifolds with spectra agreeing outside a finite set.Source used to assess the problem's recorded status.For Planar drums whose spectra differ only finitely, this source supplies a direct analogue in another geometric category without establishing planar-domain rigidity.
arXiv preprint 1005.1839, linked primary source for “Planar drums whose spectra differ only finitely,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗preprint · primary source · arXiv:1005.1839, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at abstract and transplantation construction of fully isospectral planar domains.Source used to assess the problem's recorded status.For Planar drums whose spectra differ only finitely, this source constructs exactly isospectral plane domains rather than eventual-only agreement for smooth connected domains.

An original CC0 textbook restatement motivated by the cited MathOverflow spectral question; no page wording was copied.

[#P2922] Planar drums whose spectra differ only finitely

1Context

2Problem setup

3What counts as a solution

1Status

1Records

2See also

1Lean verification

1References

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Definitions

What counts as a solution

What's known

Formal statement

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1Context

2Problem setup

3What counts as a solution

1Status

1Records

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance