It is shown by T. Kato that the Schrödinger operator with the Coulomb potential. Moreover, it is known that the Laplacian on a complete Riemannian manifold is essential self-adjoint on the space of compactly supported smooth functions. On the other hands, on a domain, the Laplacian is not essential self-adjoint. In fact, the Laplacian has at least two self-adjoint extensions: the Dirichlet Laplacian and the Neumann Laplacian. The last example suggests that an obstruction of essential self-adjointness is the "boundary" of the manifold.

In this talk, I will explain how techniques of scattering theory can be applied to judging self-adjointness of differential operators. As an application, I will present the following two results.

(1) Essential self-adjointness of real-principal type operators .

(2) Give an alternative proof of not essential self-adjointness of repulsive Schrødinger operators with a large exponent in view of scattering theory.

This is partially joint work with Shu Nakamura.