Self-adjoint extensions by additive perturbations
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 1, p. 1-20

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 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 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.

Classification:  47B25,  47B38,  47F05
@article{ASNSP_2003_5_2_1_1_0,
     author = {Posilicano, Andrea},
     title = {Self-adjoint extensions by additive perturbations},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 2},
     number = {1},
     year = {2003},
     pages = {1-20},
     zbl = {1096.47505},
     mrnumber = {1990972},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2003_5_2_1_1_0}
}
Posilicano, Andrea. Self-adjoint extensions by additive perturbations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 1, pp. 1-20. http://www.numdam.org/item/ASNSP_2003_5_2_1_1_0/

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