Drift-removing substitution for a Dirichlet Sturm-Liouville problem
= Drift-removing substitution for a Dirichlet Sturm-Liouville problem
For $\mathcal Ly=-y''-2ay'$, setting $y=e^{-ax}v$ gives $\mathcal Ly=e^{-ax}(-v''+a^2v)$. Dirichlet conditions are unchanged. The weighted self-adjoint form is $-e^{-2ax}(e^{2ax}y')'$, with weight $e^{2ax}$. On an interval of length $L$, <eigenvalues> are $(n\pi/L)^2+a^2$ and <eigenfunctions> are $e^{-ax}\sin(n\pi x/L)$.