International Journal of Science, Engineering and TechnologyAuthors: Hanumesha S T
Abstract: The quantum harmonic oscillator is one of the few exactly solvable quantum systems whose spectral and semigroup properties are completely explicit. In realistic settings, however, external fields, anharmonic interactions, lattice defects, and engineered trapping profiles introduce perturbed potentials that require rigorous operator-theoretic tools to analyze stability, self-adjointness, spectral deformation, and the validity of perturbation expansions. This paper develops a functional-analytic and operator-theoretic framework for one-dimensional harmonic oscillators with additive perturbations W(x), focusing on (i) self-adjointness via Kato-Rellich and quadratic form methods, (ii) discrete-spectrum stability and eigenvalue bounds through the minmax principle and compactness arguments, (iii) analytic perturbation theory for isolated eigenvalues and the computation of first-order energy shifts for representative perturbations, and (iv) semigroup/resolvent estimates that quantify robustness of dynamics under perturbations. In addition, we propose an uncertainty-aware parameterization of perturbed potentials using intuitionistic fuzzy sets and fuzzy graph/hypergraph abstractions, linking operator stability certificates to entropy-style and stability-style diagnostics inspired by prior fuzzy-systems work. Representative figures and tables illustrate potential profiles, spectral schematics, energy shifts, and a structured workflow connecting operator estimates to computation. The resulting manuscript provides a Word-ready, mathematics-forward template for rigorous spectral analysis of perturbed quantum oscillators while also offering practical, interpretable computational guidance.
DOI: https://doi.org/10.5281/zenodo.18068506
