Locally finite extensions and Gesztesy–Šeba realizations for the Dirac operator on a metric graph
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Hannes Gernandt
Abstract
We study extensions of direct sums of symmetric operators S = ⨁n∈ℕSn. In general there is no natural boundary triplet for S∗ even if there is one for every S∗n , n ∈ ℕ. We consider a subclass of extensions of S which can be described in terms of the boundary triplets of S∗n and investigate the self-adjointness, the semiboundedness from below and the discreteness of the spectrum. Sufficient conditions for these properties are obtained from recent results on weighted discrete Laplacians. The results are applied to Laplace and Dirac operators on metric graphs with point interactions at the vertices. In particular, we allow graphs with arbitrarily small edge length.
Abstract
We study extensions of direct sums of symmetric operators S = ⨁n∈ℕSn. In general there is no natural boundary triplet for S∗ even if there is one for every S∗n , n ∈ ℕ. We consider a subclass of extensions of S which can be described in terms of the boundary triplets of S∗n and investigate the self-adjointness, the semiboundedness from below and the discreteness of the spectrum. Sufficient conditions for these properties are obtained from recent results on weighted discrete Laplacians. The results are applied to Laplace and Dirac operators on metric graphs with point interactions at the vertices. In particular, we allow graphs with arbitrarily small edge length.
Chapters in this book
- Frontmatter I
- Preface V
- Acknowledgment VII
- Contents IX
- Elliptic differential operators with periodic and oscillating decaying coefficients 1
- Locally finite extensions and Gesztesy–Šeba realizations for the Dirac operator on a metric graph 25
- Double lacunary statistically convergent sequences of weight g 55
- Riesz probability distributions on symmetric cones 63
- A note on traces of forms with applications to the Bessel operator 75
- Pseudospectra of multivalued linear operators 85
- On the frame properties of exponentials associated to analytic families of operators and application 101
- Invariance of essential spectrum by means of generalized weak demicompactness 115
- Some results of S-pseudospectra of bounded operators in a Hilbert space 121
- Stabilization of a marine riser system by the use of viscoelastic material 131
- Sequence of linear operators converging in the generalized sense and its essential pseudospectra 147
- Left–right essential spectra of one-sided operator matrix and application 157
- Existence of Solutions for an integral equation of Chandrasekhar type in Banach algebras with respect to the weak topology 177
- Strong evolution problems in Musielak spaces 183
- Numerical solution of an axisymmetric inverse heat conduction problem 199
- Multiple solutions for nonlocal elliptic problems with concave–convex nonlinearities and weights 207
Chapters in this book
- Frontmatter I
- Preface V
- Acknowledgment VII
- Contents IX
- Elliptic differential operators with periodic and oscillating decaying coefficients 1
- Locally finite extensions and Gesztesy–Šeba realizations for the Dirac operator on a metric graph 25
- Double lacunary statistically convergent sequences of weight g 55
- Riesz probability distributions on symmetric cones 63
- A note on traces of forms with applications to the Bessel operator 75
- Pseudospectra of multivalued linear operators 85
- On the frame properties of exponentials associated to analytic families of operators and application 101
- Invariance of essential spectrum by means of generalized weak demicompactness 115
- Some results of S-pseudospectra of bounded operators in a Hilbert space 121
- Stabilization of a marine riser system by the use of viscoelastic material 131
- Sequence of linear operators converging in the generalized sense and its essential pseudospectra 147
- Left–right essential spectra of one-sided operator matrix and application 157
- Existence of Solutions for an integral equation of Chandrasekhar type in Banach algebras with respect to the weak topology 177
- Strong evolution problems in Musielak spaces 183
- Numerical solution of an axisymmetric inverse heat conduction problem 199
- Multiple solutions for nonlocal elliptic problems with concave–convex nonlinearities and weights 207
