  We study the Dirichlet problem for the weighted Schr\"odinger operator
\[-\Delta u +Vu = \lambda \rho u,\] where $\rho$ is a positive weighting
function and $V$ is a potential. Such equations appear naturally in conformal
geometry and in the composite membrane problem. Our primary goal is to
establish concavity estimates for the principle eigenfunction with respect to
conformal connections. Doing so, we obtain new bounds on the fundamental gap
problem, which is the difference between the first and second eigenvalues. In
particular, we partially resolve a conjecture of Nguyen, Stancu and Wei [IMRN
2022] on the fundamental gap of horoconvex domains. In addition, we obtain a
power convexity estimate for solutions to the torsion problem in spherical
geometry on convex domains which are not too large.
