The method of comparison equations in the solution of linear second-order differential equations (generalized W.K.B. method)

Summary

Linear second-order differential equations with known solutions are classified according to the form of the ‘comparison function’ Γ(σ)=. =U ⁻¹(σ)d² U(σ)/dσ². It is shown that the solutionsu(x) of another such equation γ(x)=u ⁻¹(x)d² u(x)/dx ² may be expressed in terms of the known solutionsU(σ) by the relationu(x)=(dσ/dx)^−1/2 U(σ) where σ(x) is given by the general fundamental equation$\gamma (x) = \left( {\frac{{d\sigma }}{{dx}}} \right)^2 \Gamma (\sigma ) + \left( {\frac{{d\sigma }}{{dx}}} \right)^{ + ^1 /_2 } \frac{{d^2 }}{{dx_2 }}\left( {\frac{{d\sigma }}{{dx}}} \right)^{ - ^1 /_2 }$

When the last (‘correction’) term in this equation can be neglected, σ(x) may be found directly; as particular cases, the ordinary W.K.B. solutions result when the comparison function chosen is Γ(σ)=1, and the Langer solutions when Γ(σ)=σ. Tables are provided of the four functions in terms of which the Langer solutions can most easily be expressed.

It is shown how the approximate solution of the general fundamental equation relating σ tox may be employed in a perturbation method which avoids the customary- and sometimes dubious—assumption that all perturbed functions may be expanded by straightforward Taylor series involving only integral powers of the perturbation parameter.

Finally, it is shown how the correction term in the general fundamental equation may be evaluated. Explicit expressions are obtained for the correction terms to the ordinary W.K.B. solutions and to the Langer solutions.