Nonconstant-$\psi $ Limit

Suppose that $p\ll Q^{1/2}$. In this limit, Equation (5.78) reduces to

$\displaystyle \frac{d}{dp}\!\left(p^2\,\frac{dY_e}{dp}\right)-[-{\rm i}\,(Q-Q_E-Q_e)]\,\frac{B(p)}{C(p)}\,p^2\,Y_e=0.$ (5.99)

In the various nonconstant-$\psi $ regimes considered in Section 5.11, the previous equation takes the form

$\displaystyle \frac{d}{dp}\!\left(p^2\,\frac{dY_e}{dp}\right)-G\,p^{k+2}\,Y_e = 0,$ (5.100)

where $k$ is real and non-negative, and $G$ is a complex constant. Let $U=p\,Y_e$. The previous equation yields

$\displaystyle \frac{d^2 U}{dp^2} - G\,p^k\,U = 0.$ (5.101)

This equation is identical in form to Equation (5.88), which we have already solved. Indeed, the solution that is bounded as $p\rightarrow\infty$ has the small-$p$ expansion (5.90), where $q=\!\sqrt{G}\,p^j/j$, $j=(k+2)/2$, and $\nu=1/(k+2)$. Matching to Equation (5.83) yields

$\displaystyle \skew{6}\hat{\mit\Delta} = -\frac{\pi\,\nu^{1-2\nu}{\mit\Gamma}(\nu)}{{\mit\Gamma}(1-\nu)}\,\,G^{-\nu}.$ (5.102)

The layer width in $p$-space again scales as $p_\ast\sim \vert G\vert^{-\nu}$. This width must be less that $Q^{1/2}$. As before, the neglect of the term involving $c_\beta$ in Equation (5.74) is justified provided that $c_\beta\ll (Q+P\,p_\ast^2)^{1/2}\,p_\ast$.