1,721,021 research outputs found
Boundary Triples and Weyl Functions for Singular Perturbations of Self-Adjoint Operators
No full textPoincaré-invariant Markov processes and Gaussian random fields on relativistic phase space
No full textSingular perturbations of abstract wave equations
No full textAbstractGiven, on the Hilbert space H0, the self-adjoint operator B and the skew-adjoint operators C1 and C2, we consider, on the Hilbert space H≃D(B)⊕H0, the skew-adjoint operator W=C21-B2C1corresponding to the abstract wave equation φ¨-(C1+C2)φ˙=-(B2+C1C2)φ. Given then an auxiliary Hilbert space h and a linear map τ:D(B2)→h with a kernel K dense in H0, we explicitly construct skew-adjoint operators WΘ on a Hilbert space HΘ≃D(B)⊕H0⊕h which coincide with W on N≃K⊕D(B). The extension parameter Θ ranges over the set of positive, bounded and injective self-adjoint operators on h.In the case C1=C2=0 our construction allows a natural definition of negative (strongly) singular perturbations AΘ of A≔-B2 such that the diagram W→WΘA→AΘis commutative
Self-adjoint extensions of restricitons
No full textWe provide a simple recipe for obtaining all self-adjoint extensions, together with their resolvent, of the symmetric operator 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 nor of the deficiency spaces of is required. Typically is a differential operator and 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 is the set of self-adjoint operators in the range of . The set of self-adjoint operators in \fh, i.e. , parametrises the relatively prime extensions. Any (\Pi,\Theta)\in \E(\fh) determines a boundary condition in the domain of the corresponding extension and explicitly appears in the formula for the resolvent . 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
On the common point spectrum of pairs of self-adjoint extensions
No full textGiven two different self-adjoint extensions of the same symmetric operator, we analyse the intersection of their point spectra. Some simple examples are provide
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