Semi-classical resolvent estimates for the Schrödinger operator on non-compact complete Riemannian manifolds. (English) Zbl 1159.58308

Cardoso, Fernando; Popov, Georgi; Vodev, Georgi

Bull. Braz. Math. Soc. (N.S.) 35, No. 3, 333-344 (2004).

Summary: We prove uniform semi-classical estimates for the resolvent of the Schrödinger operator \(h^2\Delta_g + V(x)\), \(0 < h \ll 1\), at a nontrapping energy level \(E > 0\), where \(V\) is a real-valued non-negative potential and \(\Delta_g\) denotes the positive Laplace-Beltrami operator on a non-compact complete Riemannian manifold which may have a nonempty compact smooth boundary.

MSC:

58J05	Elliptic equations on manifolds, general theory
35J10	Schrödinger operator, Schrödinger equation
31C12	Potential theory on Riemannian manifolds and other spaces
47F05	General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)

Keywords:

semi-classical resolvent estimates; non-trapping energy level; generalized geodesics