In the following question we assume the eigenvalues are sorted in descending orders and the eigen-vectors and eigen-functions are sorted accordingly.
Given a symmetric $n\times n$ matrix $P$ and a perturbed one $P'=P+H$, how could we bound the distance of $||u_i-u_i'||$ using the spectral norm $||H||$ ? Wedin or Davis-Kahan theorem bounds the distance of eigen-space, my question is how to bound the $\ell_2$ distance of eigen-vectors?
Now if P and P' are self-adjoint operators in a Hilbert space, how could we bound $||u_i-u_i'||$ if we know the operator norm of $||H||$ ? $u_i$ and $u_i'$ are eigen-functions of the operators.
Re: #2
The 1-D Schrodinger equation with a square well potential may have several bound states. As we make the well shallower (a bounded perturbation) the top one will suddenly disappear. The corresponding eigenfunction spreads out over the whole line as this point is approached and does not have a limit in $L^2$, so its changes cannot be bounded in the way you want.