On the connection of the classical and quantum mechanical completeness of a potential at infinity on complete Riemannian manifolds (Q1893624)

The author studies on a complete Riemannian \(C^\infty\)-manifold \(M\) of dimension \(n\) the Schrödinger operator \(H= -\Delta+ q(x)\), \(x\in M\), in the space \(L^2(M)\), where \(\Delta\) is the Laplace-Beltrami operator, and the potential \(q\in L^\infty_{\text{loc}}(M)\) is a real-valued measurable function. The main result: let the potential \(q\) satisfy the inequality \(q(x)\geq - Q(x)\), where \(1\leq Q(x)\leq \infty\) and \(Q^{- 1/2}\) is a Lipschitz function on the manifold \(M\). Then, if for any arbitrary piecewise-smooth curve \(l\) outgoing to infinity \(\int_l Q^{- 1/2}(x) dl= \infty\), the operator \(H|_{C^\infty_0(M)}\) is essentially self-adjoint.