Schrödinger operator with a superposition of short-range and point potentials

We study the class of Schrödinger operators whose potential terms are sums of the short-range $\boldsymbol{r}$ and point potentials. We consider the case where the short-range potential has a singularity on the support $r=0$ of the point interaction. The point interaction is constructed using the asymptotic form of the Green's function of the Schrödinger operator $-\Delta+V(\boldsymbol{r})$ with a short-range potential $V$ as $\boldsymbol{r}\to0$. We consider potentials with a singularity of the form $r^{-\rho}$, $\rho>0$, at the origin. We use the Lippmann–Schwinger integral equation in our study. We show that if the singularity of the potential is weaker than the Coulomb singularity, then the asymptotic behavior of the Green's function has a standard singularity. If the singularity of the potential has the form $r^{-\rho}$, $1\le\rho<3/2$, then an additional singularity arises in the asymptotic behavior of the Green's function. If $\rho=1$, then the additional logarithmic singularity has the same form as in the case of the Coulomb potential. If $1<\rho<3/2$, then the additional singularity has the form of the polar singularity $r^{-\rho+1}$.