Hearing the Sides: Recovering a Planar Rectangle from Eigenvalues
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915683354279936 |
|---|---|
| author | Sultanow, Eldar Hatziiliou, Andreas |
| author_facet | Sultanow, Eldar Hatziiliou, Andreas |
| contents | We present a direct, index-free method to recover the side lengths of a planar rectangle the spectrum of its Dirichelet Laplacian, assuming only access to a finite subset of eigenvalues. No modal indices $(m,n)$ are available, and the list may begin at an arbitrary unknown offset; in particular, the lowest eigenvalues may be missing, so classical formulas based on $λ_{1,0}$ and $λ_{0,1}$ cannot be used. Our reconstruction procedure extracts geometric information solely from the asymptotic density and oscillatory structure of the ordered spectrum. The area $ab$ is obtained from the high-frequency Weyl slope, while the fundamental lengths $2a$ and $2b$ appear as dominant periodic--orbit contributions in the Fourier transform of the spectral fluctuations. This separation of smooth and oscillatory components yields a robust, offset-agnostic recovery of both side lengths. The result is a fully index-free algorithm that reconstructs the geometry of a rectangular planar domain even when the spectrum is incomplete and all modal information is lost. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_23047 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hearing the Sides: Recovering a Planar Rectangle from Eigenvalues Sultanow, Eldar Hatziiliou, Andreas Spectral Theory 35P05, 35P20, 35R30, 58J50, 81Q20 G.1.8; G.1.2; I.1.2 We present a direct, index-free method to recover the side lengths of a planar rectangle the spectrum of its Dirichelet Laplacian, assuming only access to a finite subset of eigenvalues. No modal indices $(m,n)$ are available, and the list may begin at an arbitrary unknown offset; in particular, the lowest eigenvalues may be missing, so classical formulas based on $λ_{1,0}$ and $λ_{0,1}$ cannot be used. Our reconstruction procedure extracts geometric information solely from the asymptotic density and oscillatory structure of the ordered spectrum. The area $ab$ is obtained from the high-frequency Weyl slope, while the fundamental lengths $2a$ and $2b$ appear as dominant periodic--orbit contributions in the Fourier transform of the spectral fluctuations. This separation of smooth and oscillatory components yields a robust, offset-agnostic recovery of both side lengths. The result is a fully index-free algorithm that reconstructs the geometry of a rectangular planar domain even when the spectrum is incomplete and all modal information is lost. |
| title | Hearing the Sides: Recovering a Planar Rectangle from Eigenvalues |
| topic | Spectral Theory 35P05, 35P20, 35R30, 58J50, 81Q20 G.1.8; G.1.2; I.1.2 |
| url | https://arxiv.org/abs/2511.23047 |