# fa.functional analysis – Convergence on Schrödinger ground states in \$L^p\$ for \$pneq 2\$

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 $$1leq pleq infty$$.