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

Abstract

We study the class of Schrödinger operators whose potential terms are sums of the short-range V (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 −Δ+V (r) with a short-range potential V as r → 0. We consider potentials with a singularity of the form r^−ρ, ρ > 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^−ρ, 1 ≤ ρ < 3/2, then an additional singularity arises in the asymptotic behavior of the Green’s function. If ρ = 1, then the additional logarithmic singularity has the same form as in the case of the Coulomb potential. If 1 < ρ < 3/2, then the additional singularity has the form of the polar singularity r^−ρ+1.