Open perturbation and the Riccati equation: Algebraic determination of the quartic anharmonic oscillator energies and eigenfunctions

Journal of Mathematical Physics - Tập 38 Số 11 - Trang 5483-5492 - 1997
Natacha Bessis1, G. Bessis1
1Laboratoire de Physique des Lasers, U.R.A. 282 du C.N.R.S., Université Paris-Nord, avenue J. B. Clement, 93430 Villetaneuse, France

Tóm tắt

An algebraic procedure is proposed for the analytical solution of Schrödinger equations that can be viewed as a factorizable equation with an additional potential V(x). Once V(x) has been expanded in a series of suitable x-basis functions u=u(x), which are specific to each factorization type, the solution of the Riccati equation associated with the given equation is performed by means of an open perturbation technique, i.e., at each order of the perturbation, an additional balance u-dependent term is introduced so that the resulting equation becomes solvable. Since the unperturbed potential involves the whole given potential and since the balance term is expected to be small, improved results are expected at low orders of the perturbation, even at the zeroth order. The procedure, well adapted to the use of computer algebra, is applied to the solution of the gx4-anharmonic oscillator equation: by means of very simple algebraic manipulations, the trend of the exact values of the energies is rather well reproduced for a large range of values of the coupling constant (g=0.002 to g=20000).

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Tài liệu tham khảo

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Journal of Mathematical Physics

Tập 38 Số 11

5483-5492

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