Derivation of the Trace Formula: The Trace Class Result

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


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.


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    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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