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 H₀ + 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 ({R}) H - 1 (R) and H^- (1/2) + ({R}) H - (1 / 2) + δ (R) for some (0, 1/2) δ ∈ (0, 1 / 2), respectively; two-dimensional Schrödinger operators with distributional potentials in H^-1+ ({R}²) H - 1 + δ (R 2) for some (0, 1) δ ∈ (0, 1) ; and three-dimensional Schrödinger operators with distributional potentials in H^-1/2 ({R}³) H - 1 / 2 (R 3). In all cases, the Birman–Schwinger operator aligned AV () = - (H₀- I₋ℂ ({ₑⁿ) }) ^-1/2V (H₀- I₋ℂ ({ₑⁿ) }) ^-1/2, (-, 0) aligned A V (λ) = - (H 0 - λ I L