The formalism of exact 1D quantization is reviewed in detail and applied to the spectral study of three concrete Schrödinger Hamiltonians
[−d2/dq2+V(q)]± on the half-line
{q>0}, with a Dirichlet (
−) or Neumann (
+) condition at
q=0. Emphasis is put on the analytical investigation of the spectral determinants and spectral zeta functions with respect to singular perturbation parameters. We first discuss the homogeneous potential
V(q)=qN as
N→+∞ vs its (solvable)
N=∞ limit (an infinite square well): useful distinctions are established between regular and singular behaviours of spectral quantities; various identities among the square-well spectral functions are unraveled as limits of finite-
N properties. The second model is the quartic anharmonic oscillator: its zero-energy spectral determinants
det(−d2/dq2+q4+vq2)± are explicitly analyzed in detail, revealing many special values, algebraic identities between Taylor coefficients, and functional equations of a quartic type coupled to asymptotic
v→+∞ properties of Airy type. The third study addresses the potentials
V(q)=qN+vqN/2−1 of even degree: their zero-energy spectral determinants prove computable in closed form, and the generalized eigenvalue problems with
v as spectral variable admit exact quantization formulae which are perfect extensions of the harmonic oscillator case (corresponding to
N=2); these results probably reflect the presence of supersymmetric potentials in the family above.
The definition of the fluctuation terms in the matrix representations of operators, especially function operators, brings a lot of facilitations to the numerical analysis of matrix representation issues. This work considers a one‐dimensional quantum oscillator with purely quartic anharmonicity and focuses on its position and momentum expectations. By using certain level commutator operation it is possible to get two ODEs as equations of motion, which are in classical nature when the fluctuation terms are omitted. To understand how fluctuation terms affect expectation dynamical analysis we add extra temporal unknowns and construct extra ODEs over them. There seem to be an infinite ODEs chain for fluctuation altough their truncations seem to be utilizable as approximations. Here basic theory is summarized. Further details will be prasented at the conference presentation.
This is an investigation of an anharmonic oscillator characterized by the potential (ω02/ 2) q2 + λ′ q4. By using the exact relations obtained by evaluating ≪i|[H, O]∓|j > where O is an arbitrary operator, H is our total Hamiltonian and |i > and |j > are exact eigenstates of H, we derive an exact recurrence formula. This formula allows us to express τ-functions with a higher power of the variables through τ-functions with a lower power of the variables and energy eigenvalues. In this way we derive several exact relations, which are, in a sense, generalizations of the virial theorem and sum rules (Eqs. (4·5), (4·6) and (5·4)). These exact relations are the central equaions of this paper. On the basis of these exact relations we propose our “nearest neighbour level” (N. N. L) approximation, which seems to provide a good approximation scheme. We can also use our exact relations to test the validity of various approximation methods, and as an example, we discuss the “New-Tamm-Dancoff” (N. T. D)-type of approximation in detail.
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.
Quantum mechanical anharmonic oscillators and Hamiltonians for particles in external magnetic fields are related to representations of nilpotent groups. Using this connection the eigenfunctions of the quartic anharmonic oscillator with potential Vα(x)=(α+(x2/2))2 can be used to determine the eigenfunctions of a charged particle in a nonconstant magnetic field, of the form Bz=β2+β3x. The quartic anharmonic oscillator eigenvalues for low-lying states are obtained numerically and a function which interpolates between α≪0 (a double harmonic oscillator) and α≫0 (a harmonic oscillator) is shown to give a good fit to the numerical data. Approximate expressions for the quartic anharmonic oscillator eigenfunctions are then used to get the eigenfunctions for the magnetic field Hamiltonian.