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International Journal of Science Engineering and Technology

Operator-Theoretic Analysis of Quantum Harmonic Oscillators with Perturbed Potentials

Hanumesha S T

Published Apr 27, 2018• 0 citations
Publisher

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
10.5281/zenodo.18068505

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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.
This work is aimed at obtaining the energy eigenvalues for one-dimensional quantum harmonic and anharmonic oscillators perturbed by linear, quadratic, cubic and polynomial potentials. To obtain the solutions of the energy eigenvalues, we employed the time-independent perturbation theory to calculate the first and the second-order energy correction, which we used to obtain the complete generalised energy eigenvalues of the quantum harmonic oscillators with linear, quadratic, cubic and polynomial perturbation potential of the same unperturbed Hamiltonian (H).
Abstract Non-self-adjoint operators with a real discrete spectrum exhibit, in general, a quite different behavior from the one of self-adjoint operators, with the same spectrum. This fact has serious mathematical and physical implications, namely in the rigorous formulation of quantum mechanics (QM), where self-adjointness plays a central role, in particular in the spectral analysis of the involved operators. In this framework, a non-self-adjoint 2D-harmonic oscillator is considered. Its eigenvalues and eigenfunctions, as well as those of its adjoint, are explicitly determined. Although the eigensystems are complete they may not form bases, a drawback for the mathematical formulation of the QM of the system. The existence of a metric operator defining an inner product which renders the Hamiltonian self-adjoint, is discussed.
In this work we have investigated the stability and robustness of the optimal control solutions to a quantum system when the control duration goes to zero. The solutions at this limit are called “Instantaneous Solutions”. These investigations are based on the second variation of the cost functional evaluated at control solution values when the first variations of wave and costate functions are related to the first variation of external field amplitude via control equations. This form of cost functional's second variation is purely quadratic in the first variation of the external field amplitude. Investigations are conducted for an illustrative model system, one dimensional quantum harmonic oscillator under linear dipole interaction, purely quadratic objective operator in position, and purely quadratic penalty operator in momentum. We have not found the stability operator's spectrum explicitly. Instead we have employed a bound analysis to understand the system's stability and robustness. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
Abstract Given its well known spectral decomposition profile, the 1-dim harmonic oscillator potential modified by an inverse square (1-dim angular momentum-like) contribution works as an efficient platform for probing classical and quantum information quantifiers in the context of the phase-space Weyl-Wigner formalism. In particular, the phase-space informational content related to the canonical ensemble driven by such a singular oscillator can be quantified in terms of well established analytical structures. Considering that, on one hand, the singular oscillator produce a spectral decomposition profile equivalent to that one of the unmodified harmonic system – in the sense that they result into identical thermodynamic statistics, even for different statistical mixtures – on the other hand, a more complete scrutinization of their phase-space information content can capture some different aspects of the encoded information for the related quantum ensembles. Besides the identification of decoherence effects, the Wigner flow analysis is presumedly useful in identifying stable quantum configurations, according to finite temperature and interaction parameter values. Unexpectedly, our results show that the equivalence between the statistical (quantum) mechanics of the anharmonic singular oscillator and an ordinary harmonic oscillator can also be extended to the phase-space quantum purity quantifier, which is analytically computed and reproduces exactly the same quantum ensemble statistical mixture profile, which does not depend on interaction parameter values.
I Plenary talks.- An adiabatic theorem without a gap condition.- Two-dimensional periodic Pauli operator. The effective masses at the lower edge of the spectrum.- Spectral problems in the theory of photonic crystals.- Optimal eigenvalues for some Laplacians and Schroedinger operators depending on curvature.- The spectral shift operator.- On the scattering operator for the Schroedinger equation with a time-dependent potential.- H_2-construction and some applications.- Scattering with time periodic potentials and cyclic states.- Some geometry related to decay properties of the resolvent of a class of symmetric operators.- On some asymptotic formulas in the strong localization regime of the theory of disordered systems.- Spectral measures and category.- Quantum dots. A survey of rigorous results.- II Session talks.- A simple model of concentrated nonlinearity.- Anomalous electron trapping by magnetic flux tubes and electric current vortices.- On the absolutely continuous energy distribution of a quantum mechanical system in a bounded domain.- Some aspects of generalized contact interaction in one-dimensional quantum mechanics.- Traces and trace norms for semigroup differences...- About a resolvent formula.- The determinant anomaly in low-dimensional quantum systems.- Linear Boltzmann equation as the weak coupling limit of the random Schroedinger equation.- Coexistence of different spectral types for almost periodic Schroedinger equations in dimension one.- Dynamical localization for random Schroedinger operators and an application to the almost Mathieu operator.- On fractal structure of the spectrum for periodic point perturbations of the Schroedinger operator with a uniform magnetic field.- A Weyl-Berry formula for the scattering operator associated to self-similar potentials on the line.- Localization and Lifshitz tails for random quantum waveguides.- Birman-Schwinger analysis for bound states in a pair of parallel quantum waveguides with a semitransparent boundary.- On the absolute continuity of spectra of periodic elliptic operators.- Hardy inequalities for magnetic Dirichlet forms.- Adiabatic curvature, chaos and the deformation of Riemann surfaces.- Operator interpretation of resonances arising in spectral problems for 2 x 2 matrix Hamiltonians.- On the operator-norm convergence of the Trotter-Kato product formula.- A particular case of the inverse problem for the Sturm-Liouville equation with parameter dependent potential.- One-dimensional Schroedinger operators with decaying potentials.- Stability of limiting absorption under singular perturbations.- Existence of averaging integrals for self-adjoint operators.- Monotonicity versus non-monotonicity in random operators.- A model in perturbation theory.- Band gap of the spectrum in periodically curved quantum waveguides.- A list of other talks.- A list of participants.