Phase-integral approximation generated from an unspecified base function

1 Phase-integral approximation generated from an unspecified base function 

For a detailed description of this approximation we refer to Chapter 1 in Froman and Froman (1996) and to Froman and Frdman (2002). A brief description is given below. 

We introduce into (4.1) a small bookkeeping parameter A that will finally be put equal to unity. Thus we get the auxiliary differential equation 

The auxiliary differential equation (4.2) has two linearly independent solutions fi(z) and f2(z) of the form 

The choice of the unspecified base function Q(z) shows itself only in the expressions (4.6) and (4.8) for ( and q which depend explicitly on R z) and Q(z), while the functions which are expressed in terms of q and derivatives of q with respect to (, do not depend explicitly on R z) and the choice of the base function Q(z). The expressions for the functions Y m can therefore be determined once and for all. We also remark that at the zeros and poles of Q2(z) the functions Q(z) and Q 1 2(z) may have branch points, whereas the functions o, Yzn and q z)/Q z) are single-valued. Truncating the infinite series in (4.9) at n = N, we obtain 

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In the present chapter we shall start from the results obtained in Chapter 3 and treat the Stark effect of a hydrogenic atom or ion with the use of the phase-integral approximation generated from an unspecified base function developed by the present authors and briefly described in Chapter 4 of this book. Phase-integral formulas for profiles, energies and half-widths of Stark levels are obtained. The profile has a Lorentzian shape when the level is narrow but a non-Lorentzian shape when the level is broad. A formula for the half-width is derived on the assumption that the level is not too broad. 

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