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Derivation of the Trace Formula: The Trace Class Result

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

Abstract

This chapter is of a technical nature and as its center piece proves that the operator \(z\chi _{\varLambda }\mathop{\mathrm{tr}}\nolimits _{N}\big(\left (L^{{\ast}}L + z\right )^{-1} -\left (LL^{{\ast}} + z\right )^{-1}\big)\), \(z \in \varrho (-LL^{{\ast}}) \cap \varrho (-L^{{\ast}}L)\), belongs to the trace class \(\mathcal{B}_{1}\big(L^{2}(\mathbb{R}^{n})\big)\). Here \(\mathop{\mathrm{tr}}\nolimits _{N}\) describes an appropriate internal trace, and \(\chi _{\varLambda }\) is the multiplication operator of multiplying with the characteristic function of the ball centered at 0 with radius \(\varLambda> 0\) in \(\mathbb{R}^{n}\). Moreover, it is shown that the operator \(\mathop{\mathrm{tr}}\nolimits _{N}\big(\left (L^{{\ast}}L + z\right )^{-1} -\left (LL^{{\ast}} + z\right )^{-1}\big)\) vanishes in even dimension n, thus, all subsequent index computations can be confined to odd space dimensions n.

References

  1. 22.
    C. Callias, Axial anomalies and index theorems on open spaces. Commun. Math. Phys. 62, 213–234 (1978)MathSciNetCrossRefzbMATHGoogle 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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