Self-adjoint Extensions of Restrictions
By Andrea Posilicano. 2007
140 citations recorded by the source corpus.
We provide a simple recipe for obtaining all self-adjoint extensions, together with their resolvent, of the symmetric operator $S$ obtained by restricting the self-adjoint operator $A:\D(A)\subseteq\H\to\H$ to the dense, closed with respect to the graph norm, subspace $\N\subset \D(A)$. Neither the knowledge of $S^*$ nor of the deficiency spaces of $S$ is required. Typically $A$ is a differential operator and $\N$ is the kernel of some trace (restriction) operator along a null subset. We parametrise the extensions by the bundle $\pi:\E(\fh)\to\P(\fh)$, where $\P(\fh)$ denotes the set of orthogonal projections in the Hilbert space $\fh\simeq \D(A)/\N$ and $\pi^{-1}(\Pi)$ is the set of self-adjoint operators in the range of $\Pi$. The set of self-adjoint operators in $\fh$, i.e. $\pi^{-1}(1)$, parametrises the relatively prime extensions. Any $(\Pi,\Theta)\in \E(\fh)$ determines a boundary condition in the domain of the corresponding extension $A_{\Pi,\Theta}$ and explicitly appears in the formula for the resolvent $(-A_{\Pi,\Theta}+z)^{-1}$. The connection with both von Neumann's and Boundary Triples theories of self-adjoint extensions is explained. Some examples related to quantum graphs, to Schr\"odinger operators with point interactions and to elliptic boundary value problems are given.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/dc52c58619c9a92d16bc086d955bca58f85d74d2
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In Applied Mathematics
In Computational Theory and Mathematics
In Mathematical Physics
Citations in the corpus
- Boundary pairs associated with quadratic forms (2012, background): “[74] A. Posilicano, Self-adjoint extensions of restrictions, Oper.”
- A description of all self-adjoint extensions of the Laplacian and Kreĭn-type resolvent formulas on non-smooth domains (2009, methodology): “We note that in the PDE context these ideas were realized only very recently in the work of Amrein and Pearson [10], Behrndt and Langer [23], Brown, Grubb, and Wood [32], Brown, Hinchliffe, Marletta, Naboko, and Wood [33], Brown, Marletta, Naboko, and Wood [35], Gesztesy and Mitrea [55], [56], Grubb”
- Boundary Value Problems, Weyl Functions, and Differential Operators (2020, background): “However, for a selection of papers in which boundary triplet techniques were applied to differential operators and other related problems we refer to [28, 50, 86, 87, 97, 112, 113, 123, 143, 144, 155, 169, 170, 173, 174, 184, 185, 241, 288, 307, 342, 343, 344, 377, 461, 475, 477, 478, 483, 511, 529,”
- Boundary triplets and M‐functions for non‐selfadjoint operators, with applications to elliptic PDEs and block operator matrices (2007, methodology): “While we cannot undertake a comprehensive survey of the literature here, we recommend that the reader consult the works of Derkach and Malamud who developed the theory of the Weyl-M -function in the context of boundary value spaces (e.g. [10, 11]); the work of V.A. Mikhailets (e.g. the very elegant ”
- Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples (2012, background): “…and Trunk, 2007; Brasche, Malamud and Neidhardt, 2002; Brüning, Geyler and Pankrashkin, 2008; Derkach, Hassi and de Snoo, 2003; Gorbachuk and Gorbachuk, 1991; Derkach and Malamud, 1995; Karabash, Kostenko and Malamud, 2009; Kostenko and Malamud, 2010; Posilicano, 2008] and the references therein.”
- M ‐functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems (2008, result): “undary triplet with H = K = L2(Σ): (3.16) (Au,v)L 2(Ω) −(u,A ′v) L2(Ω) = (Γ1u,Γ ′ 0v)L 2(Σ) −(Γ0u,Γ ′ 1v)L 2(Σ), holds when u∈ D(A max), v∈ D(A′ ): Such reductions to L2-dualities are made in [7] and [37]. [15] did not make the modification by composition with Λ±1 2 , but worked directly with (3.12).”
- Completeness and spectral synthesis of nonselfadjoint one-dimensional perturbations of selfadjoint operators (2012, background): “We refer to [20], [5], [69], [15] and to books [30], [51], [48] for alternative treatments of singular perturbations in more general settings.”