The Hill Determinant: An Application to the Anharmonic Oscillator

Abstract

We have calculated the ground-state eigenvalues of the kx4 anharmonic oscillator nonperturbatirely, using the Hill determinant. Our results a r e in remarkable agreement with those obtained from the or el-pad6 approximants of the perturbation series.From an exhaustive numerical analysis of the integral exists for that z . To facilitate numerical perturbation series for the ground-state energy computation, Graffi et al. used Pad6 approximants level of the one-dimensional anharmonic oscillator for +(tz). for which the Hamiltonian i s given byIn this note, we wish to point out that exact values of the energy levels of the anharmonic oscillator d2can also be obtained without recourse to the stan-dard perturbation series and associated summability techniques. Our approach, essentially based Bender and Wu' have shown that the power series in h is divergent for all h though each term of the series is finite. Further, they show that the energy levels of the system orginally defined for real h > 0 can be analytically continued into the complex h plane and that the continuation has an infinite number of branch points with a limit point at h = 0. Such series a r e quite common in relativistic quantum mechanics and the usual belief i s that they a r e asymptotic in nature.' It i s well known in the mathematical literature3 that such series can often be summed4 uniquely through the use of such summability techniques a s the Stieltjes-Pad6 o r the Borel methods. Simon5 has recently investigated the anharmonic oscillator with the general anharmonic term AX'" ( m an integer > 0) and has shown that the pth energy level E:(h) i s analytic in a certain r egion of the X plane and that the perturbation series i s asymptotic to the value EF(h). In particular, he has calculated E;(A) by converting the perturbation series into a series of pad6 approximants for various values of A. In a recent communication, Graffi et uLs6 have shown how improved values of the ground-state energy level for arbitrary h can be obtained by using Pad6 approximants of the Borel transform of the asymptotic perturbation series. In essence, their method consists in r eplacing the s e r i e s C:=, anzn by the Borel sumWithin their regions of convergence both series a r e identical, but for values of z for which the series C ; = , u,zn diverges, the integral representation gives the value of the series provided the upon solving the Hill determinants in finding eigenvalues, has long been known in the literature of mathematical physics .7 From our analysis we find the following: (i) For small h (A <<O.l) the ground-state energy is just that predicted by the asymptotic perturbation series.(ii) For X 0.1 our results for the ground-state energy a r e in remarkable agreement with those obtained from the Pad6 approximants of the Borel sum of the perturbation series.(iii) The eigenvalues and eigenfunctions for excited levels can also be obtained within our framework.(iv) Our method affords a straightforward generalization to the problem with the general anharmonic term o...

Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators

“…In [13], Biswas et al use a fixed exponential factor e −x 2 /2 but subsequent studies considered e −γx 2 with γ adjustable [23,24,25] and even e −γx 2 +ρx 4 with γ and ρ adjustable [19]. Nevertheless this method may well be sketchily introduced without considering any prefactor.…”

“…Actually it is traditionally considered not as a pure power-series-based method but rather as a variational method because of an exponential prefactor usually introduced in the relation (9) between ψ and the power series (13). In [13], Biswas et al use a fixed exponential factor e −x 2 /2 but subsequent studies considered e −γx 2 with γ adjustable [23,24,25] and even e −γx 2 +ρx 4 with γ and ρ adjustable [19].…”

“…As example, with M = 150, E 0 is determined that way with an accuracy of 83 significant figures (see figure 5 and the appendix). Only the choice of the order M limits the accuracy of this estimation which is much more accurate than the previously published values [13,24,38]. An efficient and very convenient variant to the auxiliary condition (37) consists in simply imposing the vanishing of the last term of the series (35), say…”

“…One of the most popular methods used to solve the eigenvalue problem of the AO is the so-called Hill determinant method [13] which is (partly) based on a Taylor expansion of the solution in powers of the independent variable x such as (9, 13).…”

“…Section 3 presents a short review of the Taylor-seriesbased methods utilized by the past to solve the AO. Among those is the popular Hill determinant method (adapted to the AO in [13]). Although it is usually not classified that way but rather among the variational methods, it is presented here because of its close similarity with a simplistic method used sometimes to approximately solve nonlinear ODEs [7,14].…”