Symmetric pairs and self-adjoint extensions of operators, with\n applications to energy networks

2015/12/10 by Palle E. T. Jørgensen, Jorgensen, Palle E. T., Erin P. J. Pearse +1
Mathematics · Physics and Astronomy · Engineering · #Spectral Theory in Mathematical Physics #Quantum optics and atomic interactions #Molecular Junctions and Nanostructures

paper · pdf · doi:10.48550/arxiv.1512.03463

Abstract

We provide a streamlined construction of the Friedrichs extension of a\ndensely-defined self-adjoint and semibounded operator A on a Hilbert space\n\H, by means of a symmetric pair of operators. A \symmetric\npair is comprised of densely defined operators J: \H₁ \→\n\H₂ and K: \H₂ \→ \H₁ which are compatible in\na certain sense. With the appropriate definitions of \H₁ and J in\nterms of A and \H, we show that (JJ^\⋆)^-1 is the\nFriedrichs extension of A. Furthermore, we use related ideas (including the\nnotion of unbounded containment) to construct a generalization of the\nconstruction of the Krein extension of A as laid out in a previous paper of\nthe authors. These results are applied to the study of the graph Laplacian on\ninfinite networks, in relation to the Hilbert spaces \ℓ²(G) and\n\H _{mathcal E} (the energy space).\n

Related

• Self-adjoint Extensions of Restrictions
• Self-adjoint extensions of network Laplacians and applications to\n resistance metrics
• Unbounded containment in the energy space of a network and the Krein extension of the energy Laplacian
• Notes on (the Birmak-Krein-Vishik theory on) selfadjoint extensions of semibounded symmetric operators
• Self-Adjoint Operators in Extended Hilbert Spaces H⊕ W: An Application of the General GKN-EM Theorem