This is not a real answer, but there is a branch of mathematics called "semiclassical analysis" which might be related. For example, consider a degenerate version of the problem above: $$ (-h^2 \partial_x^2+V(x))u=0. $$ Here $h^2=m$ and $V=dc$; we assume that $n=1$. Then the limit as $h\to 0$ is called "semiclassical limit". What should happen (if you prescribe some boundary conditions) is that the possible solutions $u$ should get "microlocalized" near the zero set $\{p=0\}$ of the semiclassical symbol $$ p(x,\xi)=\xi^2+V(x). $$ Here a function $u$ is "microlocalized" near a subset $K$ of the cotangent bundle if a certain norm $\|Au\|$ is small for any pseudodifferential operator $A$ with symbol $a$ supported outside of $K$.
A physical explanation of the above would be that our static Schr"odinger equation governs the behaviour of a single quantum particle under the potential $V$ near the zero energy level; for small values of Planck constant $h$, this should correspond to the motion of a classical particle at this fixed energy level.
There are several sources to read about semiclassical analysis, including lecture notes by Evans-Zworski and a book by Dimassi and Sjostrand.