Suppose we want to solve the following ordinary equation $$(-\partial_x^2+W[f(x)])\psi(x)=0$$ within the range $x\in[-T,T]$ for a large number T and with the following boundary conditions $$\psi(-T)=0,\ \partial_{x}\psi(x)|_{-T}=1.$$
Now suppose for a particular function $f(x)=f_1(x)$, I have obtained a solution $\psi_1(x)$. My question is, for another function $f_2(x)$ which is very close to $f_1(x)$, what is the relation between the solution $\psi_2(x)$ and $\psi_1(x)$? Is there any mathematical theory (on ODE or operator theory) which aims to solve this kind of question?
I would really appreciate for your information. Thank you very much.