Spectral Asymmetry
The entries in Grandi's series can be paired to the eigenvalues of an infinite-dimensional operator on Hilbert space. Giving the series this interpretation gives rise to the idea of spectral asymmetry, which occurs widely in physics. The value that the series sums to depends on the asymptotic behaviour of the eigenvalues of the operator. Thus, for example, let be a sequence of both positive and negative eigenvalues. Grandi's series corresponds to the formal sum
where is the sign of the eigenvalue. The series can be given concrete values by considering various limits. For example, the heat kernel regulator leads to the sum
which, for many interesting cases, is finite for non-zero t, and converges to a finite value in the limit.
Read more about this topic: Summation Of Grandi's Series
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