LAUSR

The present paper investigates vanishing and finiteness theorems for $$L^{2{\mathcal {B}}}$$ L 2 B harmonic 1-forms on a locally conformally flat Riemannian manifold with a Schrödinger operator $${\mathcal {L}}=\varDelta +\frac{|R|}{\sqrt{n}}$$ L = Δ + | R | n . Furthermore, based on these vanishing and finiteness theorems and the theory of $$L^{2}$$ L 2 harmonic 1-forms by Li–Tam, this paper derives that the locally conformally flat Riemannian manifold with a Schrödinger operator $${\mathcal {L}}=\varDelta +\frac{|R|}{\sqrt{n}}$$ L = Δ + | R | n has one-end and finite ends. The results posed here can be regarded as a natural generalization of the work by Han (Results Math 73:54, 2018).