Using a generalized Birman–Schwinger principle developed in [31] for operators formally given by \(H_0 + V\) 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 \(H^{-1}({\mathbb {R}})\) and \(H^{-(1/2)+\delta }({\mathbb {R}})\) for some \(\delta \in (0,1/2)\) , respectively; two-dimensional Schrödinger operators with distributional potentials in \(H^{-1+\delta }({\mathbb {R}}^2)\) for some \(\delta \in (0,1)\) ; and three-dimensional Schrödinger operators with distributional potentials in \(H^{-1/2}({\mathbb {R}}^3)\) . In all cases, the Birman–Schwinger operator \(\begin{aligned} A_V(\lambda ) = -\big (H_0-\lambda I_{L^2({\mathbb {R}}^n)}\big )^{-1/2}V\big (H_0-\lambda I_{L^2({\mathbb {R}}^n)}\big )^{-1/2},\quad \lambda \in (-\infty ,0) \end{aligned}\) (here either \(H_0=-\Delta \) with \(n\in \{1,2,3\}\) or \(H_0=D_0=(-\Delta )^{1/2}\) with \(n=1\) ), is Hilbert–Schmidt with norm given explicitly by a weighted integral involving the Fourier transform of the distribution V.
