Anharmonic oscillator and the optimized basis expansion

Abstract

We introduce various optimization schemes for highly accurate calculation of the eigenvalues and the eigenfunctions of the one-dimensional anharmonic oscillators. We present several methods of analytically fixing the nonlinear variational parameter specified by the domain of the trigonometric basis functions. We show that the optimized parameter enables us to determine the energy spectrum to an arbitrary accuracy. Also, using the harmonic oscillator basis functions, we indicate that the resulting optimal frequency agrees with the one obtained by the principle of the minimal sensitivity.Comment: 15 pages, 7 figures, to appear in Applied Mathematics and Computatio

Integrability cases for the anharmonic oscillator equation

“…( 13), ( 21) and (32) of the second order differential Eqs. ( 9), ( 20) and (31), respectively, must obey the condition n < −1, thus leading to an anharmonic term of the form f 3 (t)/x n , n > 0. Such a term may be singular at x = 0.…”

“…The mathematical properties and applications of particular forms of Eq. ( 1) have been widely investigated, such as, the partial integrability of the anharmonic oscillator [2], the time-dependent driven anharmonic oscillator and its adiabaticity properties [28], toroidal p-branes, anharmonic oscillators and (hyper)elliptic solutions [29], conformal mappings and other power series methods for solving ordinary differential equations [30], and the anharmonic oscillator in the context of the optimized basis expansion [31]. The Painlevé analysis of Eq.…”

Optimized basis expansion as an extremely accurate technique for solving time-independent Schrödinger equation

“…[15]: First, the method of Ref. [15] is only applicable for the bounded power low potentials, but this method works for the general class of the bounded C ∞ potentials. Second, in our method the potentials does not have to to be symmetric.…”

“…This differential equation is not exactly solvable and for large β the behavior of the potential is very oscillatory and It is now worth mentioning the two main advantages of this technique with respect to Ref. [15]: First, the method of Ref. [15] is only applicable for the bounded power low potentials, but this method works for the general class of the bounded C ∞ potentials.…”

“…Note that, an analytic relation for this optimal length is recently presented in Ref. [15] which is only applicable for the power low potentials. A two-dimensional application of this method is also discussed in the context of quantum cosmology [16].…”