On the van der Waals interaction between a molecule and a half-infinite plate

We consider a molecule in the Born-Oppenheimer approximation interacting with a plate of infinite thickness, i.e, a half–space, which is perfectly conducting or dielectric. It is well–known in the physics literature that in this case the atom or molecule is attracted by the plate at sufficiently large distances. This effect is analogous to the well–known van der Waals interaction between neutral atoms or molecules. We prove that the interaction energy $W$ of the system is given by $W(r, v)=-C(v)r^{−3}+\mathcal{O}(r^{−4})$, where $r$ is the distance between the molecule and the plate and $v$ indicates their relative orientation. Moreover, $C(v)$ is positive and continuous, thus the atom or molecule is always pulled towards the plate at sufficiently large distances, for all relative orientations $v$. This asymptotic behavior is also well–known in the physics literature, however, we are not aware of any previous rigorous results. In addition, this is the first rigorous result identifying the leading order term in a van der Waals interaction, needing no assumptions on the multiplicity of the ground state energy of a molecule. For pedagogical reasons, we start with the case of a hydrogen atom and then we generalize the arguments to deal with a general molecule.