Discrete Spectrum and Spectral Rigidity of a Second-Order Geometric Deformation Operator

TL;DR

This paper rigorously analyzes the spectral properties of a second-order differential operator, establishing its discrete spectrum, eigenfunction completeness, and a spectral rigidity result linking spectral coefficients to a constant deformation profile.

Contribution

It provides a detailed spectral analysis of a specific geometric deformation operator, including spectrum derivation, eigenfunction properties, and a novel spectral rigidity theorem.

Findings

Discrete spectrum $oxed{C_n}$ derived

Eigenfunctions form a complete orthogonal basis

Spectral coefficients determine a constant deformation profile

Abstract

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 $C(v) = \pi$, 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.