Given a symmetric operator on a domain of some Hilbert space, there may be several extensions of it to a self-adjoint operator. These typically correspond to choices of boundary conditions (WeiJiang).
In quantum mechanics the observables are supposed to be self-adjoint operators, in particular the Hamiltonian. The physical input often directly provides only a symmetric operator, which encodes local information about the dynamics of the system . The choice of its self-adjoint extension corresponds to choices of boundary conditions on the states of the system, hence global information about the kinematics .
See the References on applications in quantum mechanics below.
Part VI, “Self-adjoint extension theory of symmetric operators” in
Konrad Schmüdgen?, Unbounded self-adjoint operators on Hilbert space, Springer GTM 265
Guangshenh Wei, Yaolin Jiang, A characterization of positive self-adjoint extensions and its application to ordinary differential operators Proceedings of the American Mathematical Society, Volume 133, Number 10 (jstor) (pdf)
An exposition and motivation by means of the simple case of a quantum particle in an infinitely deep well potential is given in
A discussion in the context of AQFT is in