Suppose I have a sequence of Schrodinger opertors $$ T_n=-\Delta +V_n $$ acting on (a subdomain of) $L^2(\mathbb{R}^d)$. Suppose that I view them as perturbations of the operator $$ T=-\Delta+V $$ where $V_n$ and $V$ are such that $T_n$ and $T$ all have compact resolvent (and for simplicity the potentials are real valued). Is there a way of bounding the difference in the eigenvalues of $T$and $T_n$ in terms of some non trivial norm of $V-V_n$, say the $L^1$ norm?
Could something be done in the less general case if all the potentials are positive?