A Krein-like Formula for Singular Perturbations of Self-Adjoint Operators and Applications

By Andrea Posilicano. 2000

143 citations recorded by the source corpus.

Abstract Given a self-adjoint operator A : D ( A )⊆ H → H and a continuous linear operator τ : D ( A )→ X with Range τ ′∩ H ′={0}, X a Banach space, we explicitly construct a family A τ Θ of self-adjoint operators such that any A τ Θ coincides with the original A on the kernel of τ . Such a family is obtained by giving a Krein-like formula where the role of the deficiency spaces is played by the dual pair ( X , X ′); the parameter Θ belongs to the space of symmetric operators from X ′ to X . When X = C one recovers the “ H −2 -construction” of Kiselev and Simon and so, to some extent, our results can be regarded as an extension of it to the infinite rank case. Considering the situation in which H = L 2 ( R n ) and τ is the trace (restriction) operator along some null subset, we give various applications to singular perturbations of non necessarily elliptic pseudo-differential operators, thus unifying and extending previously known results.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4fa719d2487bff6bb85973ccdc65c36af50a80da

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Citations in the corpus

Boundary triples and Weyl functions for singular perturbations of self-adjoint operators (2003, methodology): “Remark 3.3. The above corollary was already obtained in [17] as a direct conseguence of the results contained in [ 15 ], without making use of the theory of boundary triples.”
Boundary Value Problems, Weyl Functions, and Differential Operators (2020, background): “General operator models for highly singular perturbations involve lifting of operators in Pontryagin spaces [117, 239, 240, 241, 252, 256, 640, 641, 707].”
A description of all self-adjoint extensions of the Laplacian and Kreĭn-type resolvent formulas on non-smooth domains (2009, unknown): “13], [76], [79], [80], [81], [82], [83], [84], [87], [88], [89], [95], [100], [101], [102], [103], [104], [105], [106], [107], [108], [111], [112], [113], [114], [119], [120], [121], [127], and the references cited therein.”
Self-adjoint Extensions of Restrictions (2007, methodology): “Indeed by (2.2) and the definition of ( z) it is easy to check (see [ 25 ], Lemma 2.2, for the details) that any of such a family differs by a z-independent bounded self-adjoint operator from the family ˆ (w)(z) defined by”
Boundary relations and generalized resolvents of symmetric operators (2006, background): “The Krĕın-Naimark formula plays an important role in the extension theory of the operator A (see [1, 2, 3, 12, 17, 19, 24] and references therein) and its numerous applications to classical interpolation problems ([6, 31, 32, 33, 34, 39, 17, 19]), boundary value problems ([20, 21, 25]) and different”