is not essentially self-adjoint on the space of smooth, rapidly decaying functions. In this case, the failure of essential self-adjointness reflects a pathology in the underlying classical system: A classical particle with a formula_84 potential escapes to infinity in finite time. This operator does not have a "unique" self-adjoint, but it does admit self-adjoint extensions obtained by specifying "boundary conditions at infinity". (Since formula_3 is a real operator, it commutes with complex conjugation. Thus, the deficiency indices are automatically equal, which is the condition for having a self-adjoint extension. See the discussion of extensions of symmetric operators below.)