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 .

Part VI, “Self-adjoint extension theory of symmetric operators” in

Konrad Schmüdgen, Unbounded self-adjoint operators on Hilbert space, Springer GTM 265, 2012

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)

For applications in quantum theory

An exposition and motivation by means of the simple case of a quantum particle in an infinitely deep well potential is given in

Guy Bonneau, Jacques Faraut, Galliano Valent, Self-adjoint extensions of operators and the teaching of quantum mechanics, arXiv:quant-ph/0103153