Symmetric and self-adjoint operators

The formal definition of an unbounded self-adjoint operator in a Hilbert space sometimes is not clear for people who are not specialists in Operator Theory. The usual reason is that the definition of a symmetric
operator looks simpler and more attractive. The aim of the talk is to explain differences between these two classes of operators on simple examples and, time permitting, to show some basic applications of the spectral theory of self-adjoint operators.