Discrete Spectrum and Spectral Rigidity of a Second-Order Geometric Deformation Operator
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908430118158336 |
|---|---|
| author | Alexa, Anton |
| author_facet | Alexa, Anton |
| contents | We analyze the spectral properties of a self-adjoint second-order differential operator $\hat{C}$, defined on the Hilbert space $L^2([-v_c, v_c])$ with Dirichlet boundary conditions. We derive the discrete spectrum $\{C_n\}$, prove the completeness of the associated eigenfunctions, and establish orthogonality and normalization relations. The analysis follows the classical Sturm--Liouville framework and confirms that the deformation modes $C_n$ form a spectral basis on the compact interval. We further establish a spectral rigidity result: uniform spectral coefficients imply a constant profile $C(v) = π$, which does not belong to the Sobolev domain of the operator. These results provide a rigorous foundation for further investigations in spectral geometry and functional analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_01440 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Discrete Spectrum and Spectral Rigidity of a Second-Order Geometric Deformation Operator Alexa, Anton Spectral Theory Functional Analysis 35P05, 34L10, 47A10, 46E35 We analyze the spectral properties of a self-adjoint second-order differential operator $\hat{C}$, defined on the Hilbert space $L^2([-v_c, v_c])$ with Dirichlet boundary conditions. We derive the discrete spectrum $\{C_n\}$, prove the completeness of the associated eigenfunctions, and establish orthogonality and normalization relations. The analysis follows the classical Sturm--Liouville framework and confirms that the deformation modes $C_n$ form a spectral basis on the compact interval. We further establish a spectral rigidity result: uniform spectral coefficients imply a constant profile $C(v) = π$, which does not belong to the Sobolev domain of the operator. These results provide a rigorous foundation for further investigations in spectral geometry and functional analysis. |
| title | Discrete Spectrum and Spectral Rigidity of a Second-Order Geometric Deformation Operator |
| topic | Spectral Theory Functional Analysis 35P05, 34L10, 47A10, 46E35 |
| url | https://arxiv.org/abs/2507.01440 |