nLab self-adjoint extension




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

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.


See also Friedrichs extension.


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

A discussion in the context of local nets of observables in algebraic quantum field theory:

Applications to quantum field theory of anyons is discussed in

  • G. Amelino-Camelia, D. Bak, Schrödinger self-adjoint sxtension and quantum field theory, arXiv:hep-th/9406213
category: analysis, physics

Last revised on March 30, 2023 at 07:27:06. See the history of this page for a list of all contributions to it.