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Dirac-Type Operators

  • Fritz Gesztesy
  • Marcus Waurick
Chapter
  • 481 Downloads
Part of the Lecture Notes in Mathematics book series (LNM, volume 2157)

Abstract

This chapter is devoted to a detailed discussion of Callias’ first-order differential operator \(L = \mathcal{Q}+\varPhi\) and its associated supersymmetric Dirac-type operator \(H = \left (\begin{matrix}\scriptstyle 0 &\scriptstyle L^{{\ast}} \\ \scriptstyle L&\scriptstyle 0 \end{matrix}\right )\), with \(\mathcal{Q} =\sum _{ j=1}^{n}\gamma _{j,n}\partial _{j}\) in \(L^{2}(\mathbb{R}^{n})^{N}\) for appropriate \(N = N(n) \in \mathbb{N}\), γj, n elements of the Euclidean Dirac algebra, and Φ an appropriate potential coefficient. The center piece of this chapter is a proof of the fact that L is a densely defined, closed, Fredholm operator in \(L^{2}(\mathbb{R}^{n})^{N}\). In addition, the notion of admissible potentials Φ is introduced.

Keywords

Dirac Type Operators Admissible Potentials Potential Coefficients Fredholm Property Rellich-Kondrachov Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 2.
    R.A. Adams, J.J.F. Fournier, Sobolev spaces, 2nd edn. (Academic Press, New York, 2003)zbMATHGoogle Scholar
  2. 22.
    C. Callias, Axial anomalies and index theorems on open spaces. Commun. Math. Phys. 62, 213–234 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 52.
    F. Gesztesy, Y. Latushkin, K.A. Makarov, F. Sukochev, Y. Tomilov, The index formula and the spectral shift function for relatively trace class perturbations. Adv. Math. 227, 319–420 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 71.
    T. Kato, Perturbation Theory for Linear Operators, corr. printing of the 2nd edn. Grundlehren math. Wissensch., Vol. 274 (Springer, Berlin, 1980)Google Scholar
  5. 95.
    B. Thaller, The Dirac Equation. Texts and Monographs in Physics (Springer, Berlin, 1992)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Fritz Gesztesy
    • 1
  • Marcus Waurick
    • 2
  1. 1.Dept of MathematicsUniversity of MissouriColumbiaUSA
  2. 2.Institut für AnalysisTU DresdenDresdenGermany

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