Suppose that $H=-Delta+V$ is a Schrödinger operator with a unique ground state $psi$. Suppose that $H_n=-Delta+V_n$ is a sequence of operators such that $V_nto V$ in some sense as $ntoinfty$ (more on this in a bit), also with a unique ground state $psi_n$. I am interested in the following type of problem:

Question.Are there known sufficient conditions on the convergence of the potentials $V_nto V$ that guarantee the convergence of ground states $|psi_n-psi|_{L^p}$ in some $L^p$ norm?

Here, I am especially interested in the convergence in $L^p$ for $pneq 2$. I can find some pretty strong results in the $L^2$ setting; e.g., in this paper, for one dimension, weak convergence $langle f,V_nrangletolangle f,Vrangle$ for every bounded and continuous $f$ implies norm-resolvent convergence of the operators $H_n$ to $H$ in $L^2$. That being said, I found no result for other values of $1leq pleq infty$.