Observability Estimates for Schrödinger Operators on Eucleadian space minus tube (Recent developments in studies of resonances)

Observability estimates on Euclidean space are considered. We prove the observability estimate for Schrödinger operators on the Eucleadian set minus tube. Our proof is stable under perturbation by bounded and smooth real-valued potential. Our assumption on the operator are saisfied by some non-elliptic operators and differential operators with unbounded coefficients. Results in this article are collaboration work by the author with Fabricio Macià and Shu Nakamura