  We investigate properties of pseudodifferential operators on $L^2$ space on
manifold with ends including asymptotically conical or hyperbolic ends. Our
pseudodifferential operators are a generalization of the canonical quantization
which naturally appears in the quantum mechanics on curved spaces. We prove a
Calder\'on-Vaillancourt type theorem for our pseudodifferential operators and
discuss a construction of parametrix of elliptic differential operators on
manifolds with ends.
