3
$\begingroup$

The celebrated Kronecker limit formula gives the $\zeta$-reguralized determinant of the Laplacian on the torus $\mathbb{R}/(\mathbb{Z}\omega_1+\mathbb{Z}\omega_2)$ in terms of Dedekind eta function of $\tau=\omega_2/\omega_1$. It is proven in many texts in mathematics and physics literature.

I am looking for a reference where a version of this formula for the determinans of Laplacian acting on antiperiodic functions, i. e., the ones satisfying $f(z+m\omega_1)=(-1)^{um}f(z)$, $f(z+n\omega_2)=(-1)^{vn}f(z)$, with $u,v\in \{0,1\}$.

The computation is very similar to the $(u,v)=(0,0)$ case, but it's long-ish and I would prefer not to include it in my paper as am sure that the solution is in the literature. However, I was unable to find a clean reference. E. g., Theorem 1.2 in arXiv:1607.01566 seemingly covers this case, but it formally has a restriction $\tau\in i\mathbb{R}$. Any hints?

Improve this question
$\endgroup$
2
  • 4
    $\begingroup$ Can one reduce to the periodic case by writing the space of functions periodic of periods $(2\omega_1, \omega_2)$ as the direct sum of the spaces of periodic and "anti-periodic" functions of period $(\omega_1, \omega_2)$? $\endgroup$
    – David Loeffler
    Commented Nov 1, 2019 at 12:19
  • $\begingroup$ @DavidLoeffler, hmm, you are probably right, that should yield a short computation via period doubling formulae for theta functions. I'd still prefer a reference, but if there's none, I'll go with your suggestion. Thank you! $\endgroup$
    – Kostya_I
    Commented Nov 1, 2019 at 12:29

0

Reset to default

You must log in to answer this question.