Continuous dependence on initial conditions for second order eigenvalue problem

Question

Consider the Schrödinger eigenvalue problem in one dimension

$$\phi'' - V\phi + \mu \phi = 0$$ on $[0,a]$

with boundary $\phi(a) = c$. Suppose that I already have the existence of solution/eigenvalue. Is there some continuous dependence result on the intial value $\phi(a) = c$ that is analogous or similar to the continuous dependency theorems on problems of the form $$y'=f(x,y)$$ for nice $f$?

That is, if $c_n \to c$, then do the solutions $\phi_{c_n}$ converge to $\phi_c$?

Answer 1

Never mind. I can make it into a vector first-order ODE and prove continuous dependence as usual.