Perturbation Theory and Boundary Conditions: Analogous Treatments of Anharmonic Oscillators and Double-Wells and Similarly Related Potentials and the Calculation of Exponentially Small Contributions to Eigenvalues

1990 ◽  
Vol 38 (2) ◽  
pp. 77-164 ◽  
Author(s):  
P. Achuthan ◽  
H. J. W. Müller-Kirsten ◽  
A. Wiedemann
Keyword(s):  
Perturbation Theory ◽  
Boundary Conditions ◽  
Anharmonic Oscillators ◽  
Related Potentials ◽  
Double Wells ◽  
Exponentially Small

scholarly journals Quantum square-well with logarithmic central spike

2018 ◽  
Vol 33 (02) ◽  
pp. 1850009 ◽  
Author(s):  
Miloslav Znojil ◽  
Iveta Semorádová
Keyword(s):  
Perturbation Theory ◽  
Boundary Conditions ◽  
Closed Form ◽  
Wave Functions ◽  
Change Of Variables ◽  
State Dependent ◽  
Square Well ◽  
Strength Formula ◽  
Schrödinger Perturbation ◽  
Dependent Interaction

Singular repulsive barrier [Formula: see text] inside a square-well is interpreted and studied as a linear analog of the state-dependent interaction [Formula: see text] in nonlinear Schrödinger equation. In the linearized case, Rayleigh–Schrödinger perturbation theory is shown to provide a closed-form spectrum at sufficiently small [Formula: see text] or after an amendment of the unperturbed Hamiltonian. At any spike strength [Formula: see text], the model remains solvable numerically, by the matching of wave functions. Analytically, the singularity is shown regularized via the change of variables [Formula: see text] which interchanges the roles of the asymptotic and central boundary conditions.

scholarly journals Radial anharmonic oscillator: Perturbation theory, new semiclassical expansion, approximating eigenfunctions. II. Quartic and sextic anharmonicity cases

2020 ◽  
Vol 35 (01) ◽  
pp. 2050005
Author(s):  
J. C. del Valle ◽  
A. V. Turbiner
Keyword(s):  
Perturbation Theory ◽  
Anharmonic Oscillator ◽  
Strong Coupling Regime ◽  
Weak Coupling Regime ◽  
Fractional Powers ◽  
Anharmonic Oscillators ◽  
Relative Deviation ◽  
Variational Calculations ◽  
Compact Approximation ◽  
Semiclassical Expansion

In our previous paper I (del Valle–Turbiner, 2019) a formalism was developed to study the general [Formula: see text]-dimensional radial anharmonic oscillator with potential [Formula: see text]. It was based on the Perturbation Theory (PT) in powers of [Formula: see text] (weak coupling regime) and in inverse, fractional powers of [Formula: see text] (strong coupling regime) in both [Formula: see text]-space and in [Formula: see text]-space, respectively. As a result, the Approximant was introduced — a locally-accurate uniform compact approximation of a wave function. If taken as a trial function in variational calculations, it has led to variational energies of unprecedented accuracy for cubic anharmonic oscillator. In this paper, the formalism is applied to both quartic and sextic, spherically-symmetric radial anharmonic oscillators with two term potentials [Formula: see text], [Formula: see text], respectively. It is shown that a two-parametric Approximant for quartic oscillator and a five-parametric one for sextic oscillator for the first four eigenstates used to calculate the variational energy are accurate in 8–12 figures for any [Formula: see text] and [Formula: see text], while the relative deviation of the Approximant from the exact eigenfunction is less than [Formula: see text] for any [Formula: see text].

Effect of the boundary conditions on the accuracy of perturbation theory

1987 ◽  
Vol 55 (10) ◽  
pp. 924-929
Author(s):  
M. Coronado ◽  
N. Domínguez ◽  
J. Flores ◽  
C. de la Portilla
Keyword(s):  
Perturbation Theory ◽  
Boundary Conditions

Large-order behavior of the convergent perturbation theory for anharmonic oscillators

1999 ◽  
Vol 59 (1) ◽  
pp. 102-106 ◽  
Author(s):  
L. Skála ◽  
J. Čížek ◽  
E. J. Weniger ◽  
J. Zamastil
Keyword(s):  
Perturbation Theory ◽  
Anharmonic Oscillators ◽  
Large Order
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