Dirac-Type Operators

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


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.


Dirac Type Operators Admissible Potentials Potential Coefficients Fredholm Property Rellich-Kondrachov Theorem 
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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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