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Setup: The Time Independent Schrodinger Equation(Eigenvalue problem):

$(-\frac{\hbar^2}{2m}\Delta +V)\psi = E\psi$

(On any 'reasonable' domain $D \subset \mathbb{R^3}$ with 'reasonable' boundary conditions($\psi \in L^2(D)$ atleast) with $V: D \rightarrow \mathbb{R}$ a real valued potential function, again can roughly assume it decays at large distances, but can have isolated singularities; can reduce down to a $1$D problem to simplify it as well)

When dealing with computing bound states/bound state energies of say an electron in complicated potentials, one trick I have seen informally is that we can replace the given complicated potential with something that has 'similar behaviour' but has known eigenstates/eigenvalues to 'estimate' the eigenstates/eigenvalues. What I wish to know is, under what conditions is something like this scheme faithful/what conditions need to be imposed to achieve this kind of approximation? Roughly speaking, If $V$ is the given potential, and $V_{\epsilon}$ is another potential whose eigenvalues and eigenstates are known, such that $\|V_\epsilon-V\|<\epsilon$(in some norm) then if $E_{\epsilon}, E$ denote the ground state energies(the lowest energy eigenvalues) in $V_{\epsilon}, V$ respectively, then what is an estimate on $|E_{\epsilon}-E|$? Is it of the order $\epsilon$?

I do not know what this class of problems is called so I could not look for it by just googling for example. Any references on it are requested.