Birman–Schwinger Principles for Schrödinger Operators with Distributional Potentials Revisited

Abstract Using a generalized Birman–Schwinger principle developed in [31] for operators formally given by $$H_0 + V$$ H 0 + V and the theory of Sobolev multipliers, we develop Birman–Schwinger principles for the following concrete situations: one-dimensional Schrödinger and massless relativistic Schrödinger operators with distributional potentials from $$H^{-1}({\mathbb {R}})$$ H - 1 ( R ) and $$H^{-(1/2)+\delta }({\mathbb {R}})$$ H - ( 1 / 2 ) + δ ( R ) for some $$\delta \in (0,1/2)$$ δ ∈ ( 0 , 1 / 2 ) , respectively; two-dimensional Schrödinger operators with distributional potentials in $$H^{-1+\delta }({\mathbb {R}}^2)$$ H - 1 + δ ( R 2 ) for some $$\delta \in (0,1)$$ δ ∈ ( 0 , 1 ) ; and three-dimensional Schrödinger operators with distributional potentials in $$H^{-1/2}({\mathbb {R}}^3)$$ H - 1 / 2 ( R 3 ) . In all cases, the Birman–Schwinger operator $$\begin{aligned} A_V(\lambda ) = -\big (H_0-\lambda I_{L^2({\mathbb {R}}^n)}\big )^{-1/2}V\big (H_0-\lambda I_{L^2({\mathbb {R}}^n)}\big )^{-1/2},\quad \lambda \in (-\infty ,0) \end{aligned}$$ A V ( λ ) = - ( H 0 - λ I L