Essential Self-Adjointness of the Geometric Deformation Operator on a Compact Interval

TL;DR

This paper proves that a specific second-order differential operator derived from a smooth deformation function on a compact interval is essentially self-adjoint, ensuring its mathematical consistency for spectral analysis.

Contribution

The paper explicitly verifies the essential self-adjointness of the geometric deformation operator on a compact interval, including symmetry and deficiency index calculations.

Findings

Operator is symmetric and essentially self-adjoint

Deficiency indices vanish, confirming self-adjointness

Ensures mathematical consistency for spectral analysis

Abstract

We define a second-order differential operator $\hat{C}$ on the Hilbert space $L^2([-v_c, v_c])$, constructed from a smooth deformation function $C(v)$. The operator is considered on the Sobolev domain $H^2([-v_c, v_c]) \cap H^1_0([-v_c, v_c])$ with Dirichlet boundary conditions. We prove that $\hat{C}$ is essentially self-adjoint by verifying its symmetry and computing von Neumann deficiency indices, which vanish. All steps are carried out explicitly. This result ensures the mathematical consistency of the operator and enables future spectral analysis on compact intervals.