Russian Physics Journal

, Volume 51, Issue 2, pp 115–157 | Cite as

Constructing quantum observables and self-adjoint extensions of symmetric operators. III. Self-adjoint boundary conditions

  • B. L. Voronov
  • D. M. Gitman
  • I. V. Tyutin
Elementary Particle Physics and Field Theory


This paper completes the review of the theory of self-adjoint extensions of symmetric operators for physicists as a basis for constructing quantum-mechanical observables. It contains a comparative presentation of the well-known methods and a newly proposed method for constructing ordinary self-adjoint differential operators associated with self-adjoint differential expressions in terms of self-adjoint boundary conditions. The new method has the advantage that it does not require explicitly evaluating deficient subspaces and deficiency indices (these latter are determined in passing) and that boundary conditions are of explicit character irrespective of the singularity of a differential expression. General assertions and constructions are illustrated by examples of well-known quantum-mechanical operators like momentum and Hamiltonian.


Adjoint Operator Symmetric Operator Boundary Form Natural Domain Deficiency Index 
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Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.P. N. Lebedev Institute of Physics of the Russian Academy of SciencesMoscowRussia
  2. 2.São Paulo UniversitySão PauloBrasil

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