# P2922: Planar drums whose spectra differ only finitely

- ID: `P2922`
- Reference: `planar-drums-finitely-different-spectra`
- Page: https://theoremdb.org/statements/P2922
- Record maturity: Reviewed problem with recorded work

## 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?

### Problem setup

- **Definition.** The Dirichlet eigenvalues are the eigenvalues of -Delta on a domain with zero boundary values, repeated according to multiplicity.
- **Remark.** The full spectra differ when lambda_j(Omega_1) differs from lambda_j(Omega_2) for at least one index j<N.

### What 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.

## Status

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](#reference-3) [2](#reference-2) [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

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.

- 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.
- The MathOverflow page and every visible answer and comment were checked on 2026-07-27. No argument there decides whether a finite initial spectral discrepancy can occur for smooth planar domains.
- 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.

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

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `planar-drums-finitely-different-spectra`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>MathOverflow question 474950, “Planar drums whose spectra differ only finitely,” checked 2026-08-01. Question 474950 and every visible answer and comment were checked on 2026-07-27. https://mathoverflow.net/questions/474950/can-two-drums-almost-sound-the-same
   - 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.
   - forum; reference source; checked 2026-08-01
   - Source use: citation_only
   - 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.
2. <a id="reference-2"></a>Sudhir Pujahari and Punya Plaban Satpathy, “Near Isospectrality and Spectral Rigidity for Compact Locally Symmetric Manifolds,” arXiv:2606.09320 (2026). abstract and spectral-rigidity theorem for compact locally symmetric manifolds with spectra agreeing outside a finite set https://arxiv.org/abs/2606.09320
   - preprint; reference source; arXiv:2606.09320, checked 2026-08-01; checked 2026-08-01
   - Source use: citation_only
   - 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.
3. <a id="reference-3"></a>Peter Buser, John Conway, Peter Doyle, and Klaus-Dieter Semmler, “Some planar isospectral domains,” arXiv:1005.1839 (2010). abstract and transplantation construction of fully isospectral planar domains https://arxiv.org/abs/1005.1839
   - preprint; reference source; arXiv:1005.1839, checked 2026-08-01; checked 2026-08-01
   - Source use: citation_only
   - 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.