Derivation of the Trace Formula: Diagonal Estimates

Abstract

This chapter is a direct continuation of the preceding one and computes the actual trace of the operator $\chi _{\varLambda }B_{L}(z) = z\chi _{\varLambda }\mathop{\mathrm{tr}}\nolimits _{N}\big(\left (L^{{\ast}}L + z\right )^{-1} -\left (LL^{{\ast}} + z\right )^{-1}\big)$ for odd space dimensions n. An application of Montel’s theorem plays a decisive role in this trace computation, in addition it should be noted that the case n = 3 is more subtle than $n\geqslant 5$ and requires special attention to follow in Chap. 9