| Contents |
| Essential self-adjointness of an operators $H$ has immediate applications in probability and
physics. Indeed, in general an operator $H$ has several self-adjoint extensions $H'$.
This yields Markov processes with transition semigroups $p_t = e^{-tH'}.$
The essential self-adjointness of $H$ implies that there is only one extension $H_F$: the Frederich extension.
We therefore have a unique such semigroup and thus a unique Markov process with generator $H_F$.
On the other hand, the essential self-adjointness of $H$, has the analytic consequence of the uniqueness of the quantum dynamics defined by $H$.
We will talk about necessary and sufficient conditions for Essentially Self-adjointness of a class of Relativistic Schr\"odinger Operators with, as a particular case, the Coulomb potential. |
|