Self-adjoint extensions by additive perturbations

Posilicano, Andrea

Posilicano, Andrea

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 1, pp. 1-20.

Résumé

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 $D (A_{Θ}) \cap D (A) = 𝒩$ can be additively decomposed by the sum $A_{Θ} = \bar{A} + T_{Θ}$ , where both the operators $\bar{A}$ and $T_{Θ}$ take values in the strong dual of $D (A)$ . The operator $\bar{A}$ 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.

MR Zbl

Classification : 47B25, 47B38, 47F05

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     author = {Posilicano, Andrea},
     title = {Self-adjoint extensions by additive perturbations},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {1--20},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 2},
     number = {1},
     year = {2003},
     mrnumber = {1990972},
     zbl = {1096.47505},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2003_5_2_1_1_0/}
}

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Posilicano, Andrea. Self-adjoint extensions by additive perturbations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 1, pp. 1-20. http://www.numdam.org/item/ASNSP_2003_5_2_1_1_0/

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