Spectral Stability and Evolution Generated by Self-Adjoint Operators

This work studies spectral stability and evolution generated by self-adjoint operators on Hilbert spaces. Under the assumption of a positive spectral gap, exponential decay of the associated semigroup follows from the spectral theorem. The framework connects gradient-flow evolution, spectral decomposition, and heat kernel representations. Geometric realizations arise from Schrödinger-type operators on compact Riemannian manifolds involving the Laplace–Beltrami operator. Discrete recurrence laws can also be interpreted as finite-dimensional analogues of spectral evolution. The results highlight a common operator structure underlying stability phenomena in spectral geometry, partial differential equations, and mathematical physics.This manuscript is currently under review at the Journal of Geometry and Physics.Andrew Kim (2026). Spectral Stability and Evolution Generated by Self-Adjoint Operators. Zenodo. https://doi.org/10.5281/zenodo.18973572