High-precision numerical determination of eigenvalues for a double-well potential related to the Zinn-Justin conjecture

Abstract

A numerical method of high precision is used to calculate the energy eigenvalues and eigenfunctions for a symmetric double-well potential. The method is based on enclosing the system within two infinite walls with a large but finite separation and developing a power series solution for the Schrödinger equation. The obtained numerical results are compared with those obtained on the basis of the Zinn-Justin conjecture and found to be in an excellent agreement.

Spectrum of one-dimensional anharmonic oscillators

“…The power-series method has been also applied to the case of one dimensional multi-well oscillator [12]. In addition, it has been used in [13] to justify numerical results based on the Zinn-Justin conjecture [14]. We show in work under preparation that the power-series method can be extended to the case of three-dimensional spherically symmetric potentials.…”

Quasi-exact solutions for two interacting electrons in two-dimensional anisotropic dots

“…The energy eigenvalues of the system have been obtained numerically as zeros of a function, calculated from its power series representation. Moreover, it has been shown that the bound-state energies of the confined system approach rapidly those of the unconfined oscillator for increasing R [13,14,15]. Here we show that solving the two separated equations (12) with the help of the power series representation (15) simultaneously with the condition β 1 + β 2 = 2 enables us to determine effectively the energy levels of the QD to very high precision.…”

Eigenvalues from power-series expansions: an alternative approach

“…Table 1 shows the results of the Hankel-Padé calculation of the ground-state eigenvalue of the Schrödinger equation ( 1) with the rational potential (10) for λ = 1 and three values of g. Notice that present Hankel-Padé results are more accurate than those obtained earlier by means of the Rayleigh-Ritz variational method [14],…”

“…Among the approaches applied to this model we mention perturbation theory [15,22,24,30,52], including the 1/N expansion [38,42,43], variational methods [14,16,34,55,57], and in particular the Rayleigh-Ritz method [14,34,55,57]. One can easily obtain exact solutions to the Schrödinger equation with the potential (10) for some values of the parameters λ and g [17,[19][20][21][22][23][25][26][27][28]35,37,39,41,[43][44][45][46][47][48][49]51,57] that prove suitable for testing approximate methods.…”

“…Power-series methods have proved to yield remarkably accurate eigenvalues of simple one-dimensional and central-field quantum-mechanical models [1][2][3][4][5][6][7][8][9][10][11]. There are basically two different approaches: on the one hand, the use of Dirichlet boundary conditions at the endpoints of a sufficiently wide interval [4,10], on the other, the Hilldeterminant method and its variants [1][2][3][4][5][6][7][8][9]11]. In this paper we focus our attention on the latter that has been applied to a wide variety of problems, including the vibrationrotation spectra of diatomic molecules [5,6,8].…”