Let one-dimentional Sturm–Liouville problem: $$ \frac{d^2f(x)}{dx^2}+U(x)f(x)=\lambda f(x) $$ with some appropriate boundary conditions, have a set of solutions $\{\lambda_i,\phi_i\}$.
Let further the sequence of functions $u^{(n)}(x)$ uniformly converge to $U(x)$ as $n$ tends to infinity. Is it true, that assuming the same boundary conditions the set of solutions of the approximate problem $$ \frac{d^2f(x)}{dx^2}+u^{(n)}(x)f(x)=\lambda f(x), $$ $\{\lambda^{(n)}_i,\phi^{(n)}_i\}$ converges to $\{\lambda_i,\phi_i\}$?