Separated solutions of the Black-Scholes equation ($V(S,t)=S^{\alpha }{e}^{(r-{\lambda }_{\alpha })t},{\lambda }_{\alpha }=\frac{1}{2}{\sigma }^2\alpha (\alpha -1)+(r-q)\alpha$)

In the Black-Scholes equation with continuous stock dividends, separation of variables gives $g^{\prime }=(r-\lambda )g$ and ${\sigma }^2{S}^2{f}^{\prime \prime }/2+(r-q)S{f}^{\prime }-\lambda f=0$. Substituting $f=S^{\alpha }$ gives the displayed eigenvalue. A stationary solution uses the two roots of ${\sigma }^2\alpha (\alpha -1)/2+(r-q)\alpha -r=0$. For positive volatility and $r>0$, their product is negative, so they are distinct real roots.