Analysis of Schr\"odinger operators with inverse square potentials I: regularity results in 3D

Let $V$ be a potential on ${\text{RR}}^3$ that is smooth everywhere except at a discrete set $\text{maS}$ of points, where it has singularities of the form $Z\mathrm{/}{\rho }^2$, with $\rho (x)=\mathrm{\mid }x-p\mathrm{\mid }$ for $x$ close to $p$ and $Z$ continuous on ${\text{RR}}^3$ with $Z(p)>-1\mathrm{/}4$ for $p\in \text{maS}$. Also assume that $\rho$ and $Z$ are smooth outside $\text{maS}$ and $Z$ is smooth in polar coordinates around each singular point. We either assume that $V$ is periodic or that the set $\text{maS}$ is finite and $V$ extends to a smooth function on the radial compactification of ${\text{RR}}^3$ that is bounded outside a compact set containing $\text{maS}$. In the periodic case, we let $\mathrm{\Lambda }$ be the periodicity lattice and define $\text{TT}:={\text{RR}}^3\mathrm{/}\mathrm{\Lambda }$. We obtain regularity results in weighted Sobolev space for the eigenfunctions of the Schr"odinger-type operator $H=-\mathrm{\Delta }+V$ acting on $L^2(\text{TT})$, as well as for the induced $\text{vt}k$–Hamiltonians $\text{Hk}$ obtained by restricting the action of $H$ to Bloch waves. Under some additional assumptions, we extend these regularity and solvability results to the non-periodic case. We sketch some applications to approximation of eigenfunctions and eigenvalues that will be studied in more detail in a second paper.