On the M.Kac problem with augmented data
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911888384720896 |
|---|---|
| author | Belishev, M. I. Vakulenko, A. F. |
| author_facet | Belishev, M. I. Vakulenko, A. F. |
| contents | Let $Ω$ be a bounded plane domain. As is known, the spectrum $0<λ_1<λ_2\leqslant\dots$ of its Dirichlet Laplacian $L=-Δ{\upharpoonright}[H^2(Ω)\cap H^1_0(Ω)]$ does not determine $Ω$ (up to isometry). By this, a reasonable version of the M.Kac problem is to augment the spectrum with relevant data that provide the determination.
To give the spectrum is to represent $L$ in the form $\tilde L=ΦLΦ^*={\rm diag\,}\{λ_1,λ_2,\dots\}$ in the space ${\bf l}_2$, where $Φ:L_2(Ω)\to{\bf l}_2$ is the Fourier transform. Let ${\mathscr K}=\{h\in L_2(Ω)\,|\,\,Δh=0\,\,{\rm in}\,\,Ω\}$ be the harmonic function subspace, $\tilde{\mathscr K}=Φ{\mathscr K}\subset{\bf l}_2$. We show that, in a generic case, the pair $\tilde L,\tilde {\mathscr K}$ determines $Ω$ up to isometry, what holds not only for the plain domains (drums) but for the compact Riemannian manifolds of arbitrary dimension, metric, and topology. Thus, the subspace $\tilde{\mathscr K}\subset{\bf l}_2$ augments the spectrum, making the problem uniquely solvable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_16629 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the M.Kac problem with augmented data Belishev, M. I. Vakulenko, A. F. Mathematical Physics 47Axx, 47B25, 35R30 Let $Ω$ be a bounded plane domain. As is known, the spectrum $0<λ_1<λ_2\leqslant\dots$ of its Dirichlet Laplacian $L=-Δ{\upharpoonright}[H^2(Ω)\cap H^1_0(Ω)]$ does not determine $Ω$ (up to isometry). By this, a reasonable version of the M.Kac problem is to augment the spectrum with relevant data that provide the determination. To give the spectrum is to represent $L$ in the form $\tilde L=ΦLΦ^*={\rm diag\,}\{λ_1,λ_2,\dots\}$ in the space ${\bf l}_2$, where $Φ:L_2(Ω)\to{\bf l}_2$ is the Fourier transform. Let ${\mathscr K}=\{h\in L_2(Ω)\,|\,\,Δh=0\,\,{\rm in}\,\,Ω\}$ be the harmonic function subspace, $\tilde{\mathscr K}=Φ{\mathscr K}\subset{\bf l}_2$. We show that, in a generic case, the pair $\tilde L,\tilde {\mathscr K}$ determines $Ω$ up to isometry, what holds not only for the plain domains (drums) but for the compact Riemannian manifolds of arbitrary dimension, metric, and topology. Thus, the subspace $\tilde{\mathscr K}\subset{\bf l}_2$ augments the spectrum, making the problem uniquely solvable. |
| title | On the M.Kac problem with augmented data |
| topic | Mathematical Physics 47Axx, 47B25, 35R30 |
| url | https://arxiv.org/abs/2405.16629 |