Self-adjoint extensions by additive perturbations

Andrea Posilicano

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2003)

  • Volume: 2, Issue: 1, page 1-20
  • ISSN: 0391-173X

Abstract

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Let A 𝒩 be the symmetric operator given by the restriction of A to 𝒩 , where A is a self-adjoint operator on the Hilbert space ℋ and 𝒩 is a linear dense set which is closed with respect to the graph norm on D ( A ) , the operator domain of A . We show that any self-adjoint extension A Θ of A 𝒩 such that D ( A Θ ) ∩ D ( A ) = 𝒩 can be additively decomposed by the sum A Θ = A ¯ + T Θ , where both the operators A ¯ and T Θ take values in the strong dual of D ( A ) . The operator A ¯ is the closed extension of A to the whole ℋ whereas T Θ is explicitly written in terms of a (abstract) boundary condition depending on 𝒩 and on the extension parameter Θ , a self-adjoint operator on an auxiliary Hilbert space isomorphic (as a set) to the deficiency spaces of A 𝒩 . The explicit connection with both Kreĭn’s resolvent formula and von Neumann’s theory of self-adjoint extensions is given.

How to cite

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Posilicano, Andrea. "Self-adjoint extensions by additive perturbations." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 2.1 (2003): 1-20. <http://eudml.org/doc/84497>.

@article{Posilicano2003,
abstract = {Let $A_\mathcal \{N\}$ be the symmetric operator given by the restriction of $A$ to $\mathcal \{N\}$, where $A$ is a self-adjoint operator on the Hilbert space $\mathcal \{H\}$ and $\mathcal \{N\}$ is a linear dense set which is closed with respect to the graph norm on $D(A)$, the operator domain of $A$. We show that any self-adjoint extension $A_\Theta $ of $A_\mathcal \{N\}$ such that $D(A_\Theta )\cap D(A)=\mathcal \{N\}$ can be additively decomposed by the sum $\,A_\Theta \,=\,\bar\{A\}+T_\Theta $, where both the operators $\bar\{A\}$ and $T_\Theta $ take values in the strong dual of $D(A)$. The operator $\bar\{A\}$ is the closed extension of $A$ to the whole $\mathcal \{H\}$ whereas $T_\Theta $ is explicitly written in terms of a (abstract) boundary condition depending on $\mathcal \{N\}$ and on the extension parameter $\Theta $, a self-adjoint operator on an auxiliary Hilbert space isomorphic (as a set) to the deficiency spaces of $A_\mathcal \{N\}$. The explicit connection with both Kreĭn’s resolvent formula and von Neumann’s theory of self-adjoint extensions is given.},
author = {Posilicano, Andrea},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {self-adjoint extensions of a symmetric operator; Krein-type resolvent formula},
language = {eng},
number = {1},
pages = {1-20},
publisher = {Scuola normale superiore},
title = {Self-adjoint extensions by additive perturbations},
url = {http://eudml.org/doc/84497},
volume = {2},
year = {2003},
}

TY - JOUR
AU - Posilicano, Andrea
TI - Self-adjoint extensions by additive perturbations
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2003
PB - Scuola normale superiore
VL - 2
IS - 1
SP - 1
EP - 20
AB - Let $A_\mathcal {N}$ be the symmetric operator given by the restriction of $A$ to $\mathcal {N}$, where $A$ is a self-adjoint operator on the Hilbert space $\mathcal {H}$ and $\mathcal {N}$ is a linear dense set which is closed with respect to the graph norm on $D(A)$, the operator domain of $A$. We show that any self-adjoint extension $A_\Theta $ of $A_\mathcal {N}$ such that $D(A_\Theta )\cap D(A)=\mathcal {N}$ can be additively decomposed by the sum $\,A_\Theta \,=\,\bar{A}+T_\Theta $, where both the operators $\bar{A}$ and $T_\Theta $ take values in the strong dual of $D(A)$. The operator $\bar{A}$ is the closed extension of $A$ to the whole $\mathcal {H}$ whereas $T_\Theta $ is explicitly written in terms of a (abstract) boundary condition depending on $\mathcal {N}$ and on the extension parameter $\Theta $, a self-adjoint operator on an auxiliary Hilbert space isomorphic (as a set) to the deficiency spaces of $A_\mathcal {N}$. The explicit connection with both Kreĭn’s resolvent formula and von Neumann’s theory of self-adjoint extensions is given.
LA - eng
KW - self-adjoint extensions of a symmetric operator; Krein-type resolvent formula
UR - http://eudml.org/doc/84497
ER -

References

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