On Self-Adjoint Extensions and Symmetries in Quantum Mechanics

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• 作者：Alberto Ibort ; Fernando Lledó ; Juan Manuel Pérez-Pardo
• 刊名：Annales Henri Poincare
• 出版年：2015
• 出版时间：October 2015
• 年：2015
• 卷：16
• 期：10
• 页码：2367-2397
• 全文大小：711 KB
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• 作者单位：Alberto Ibort (1) (2)
  Fernando Lledó (1) (2)
  Juan Manuel Pérez-Pardo (1) (2) (3)
  
  1. Department of Mathematics, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911, Leganés (Madrid), Spain
  2. Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), c./ Nicolás Cabrera 13-15, Campus de Cantoblanco, UAM, 28049, Madrid, Spain
  3. INFN-Sezione di Napoli, Via Cintia Edificio 6, 80126, Naples, Italy
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Mathematical and Computational Physics
  Dynamical Systems and Ergodic Theory
  Quantum Physics
  Mathematical Methods in Physics
  Relativity and Cosmology
  Elementary Particles and Quantum Field Theory
  
• 出版者：Birkh盲user Basel
• ISSN：1424-0661

文摘

Given a unitary representation of a Lie group G on a Hilbert space \({\mathcal H}\), we develop the theory of G-invariant self-adjoint extensions of symmetric operators using both von Neumann’s theorem and the theory of quadratic forms. We also analyze the relation between the reduction theory of the unitary representation and the reduction of the G-invariant unbounded operator. We also prove a G-invariant version of the representation theorem for closed and semi-bounded quadratic forms. The previous results are applied to the study of G-invariant self-adjoint extensions of the Laplace–Beltrami operator on a smooth Riemannian manifold with boundary on which the group G acts. These extensions are labeled by admissible unitaries U acting on the L 2-space at the boundary and having spectral gap at ?. It is shown that if the unitary representation V of the symmetry group G is traceable, then the self-adjoint extension of the Laplace–Beltrami operator determined by U is G-invariant if U and V commute at the boundary. Various significant examples are discussed at the end. Communicated by Karl-Henning Rehren.A. Ibort and J. M. Pérez-Pardo are partly supported by the project MTM2010-21186-C02-02 of the spanish Ministerio de Ciencia e Innovación and QUITEMAD programme P2009 ESP-1594. F. Lledó was partially supported by projects DGI MICIIN MTM2012-36372-C03-01 and Severo Ochoa SEV-2011-0087 of the spanish Ministry of Economy and Competition. J. M. Pérez-Pardo was also partially supported in 2011 and 2012 by mobility grants of the -em class="EmphasisTypeItalic ">Universidad Carlos III de Madrid-