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Published in:
2017 | OriginalPaper | Chapter
8. Numerical Results for Spectra and Traces of the Transfer Operator for Character Deformations
Author : Markus Szymon Fraczek
Published in: Selberg Zeta Functions and Transfer Operators
Publisher: Springer International Publishing
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Abstract
In Chap. 7 we discussed how to evaluate the Selberg zeta function \(Z^{\left (n\right )}(\beta,\chi )\) by computing the spectrum of the transfer operators To obtain a numerical approximation of the spectrum of the transfer operator \(\mathcal{L}_{\beta,\varepsilon,\chi }^{\left (n\right )}\) in Proposition 7.5 we approximate this operator by the matrix \(\mathcal{M}_{\beta,\varepsilon,\chi }^{\left (n\right ),N}\) in Proposition 7.7 and compute its spectrum.
$$\displaystyle{\tilde{\mathcal{L}}_{\beta,\chi }^{\left (n\right )} = \left (\begin{array}{cc} 0 &\mathcal{L}_{\beta,+1,\chi }^{\left (n\right )} \\ \mathcal{L}_{\beta,-1,\chi }^{\left (n\right )} & 0 \end{array} \right ),\mathcal{L}_{\beta,+1,\chi }^{\left (n\right )}\mathcal{L}_{\beta,-1,\chi }^{\left (n\right )}\text{ and }\mathcal{P}_{ k}\mathcal{L}_{\beta,+1,\chi }^{\left (n\right )}.}$$
$$\displaystyle\begin{array}{rcl} \left [\mathcal{L}_{\beta,\varepsilon,\chi }^{\left (n\right )}\vec{f}\left (z\right )\right ]_{ i}& =& \sum _{k=0}^{\infty }\sum _{ s=0}^{\infty }\sum _{ m=1}^{n}\sum _{ j=1}^{\mu _{n} }\left [U^{\chi }\left (ST^{m\varepsilon }\right )\right ]_{ i,j}\frac{f_{j}^{\left (k\right )}\left (1\right )} {k!} \sum _{t=0}^{k}\binom{k}{t}\frac{\left (-1\right )^{k-t+s}} {n^{2\beta +t+s}} {}\\ & & \frac{1} {s!} \frac{\varGamma (2\beta +t+s)} {\varGamma (2\beta + t)} \varPhi \!\left (\chi \left (r_{j}^{\left (n\right )}T^{n\varepsilon }\left (r_{ j}^{\left (n\right )}\right )^{-1}\right ),2\beta +t+s, \frac{m+1} {n} \right )\!\left (z-1\right )^{s}, {}\\ \end{array}$$
$$\displaystyle\begin{array}{rcl} \left [\left (\mathcal{M}_{\beta,\varepsilon,\chi }^{\left (n\right ),N}\right )_{ s,k}\right ]_{i,j}& =& \frac{1} {s!}\sum _{t=0}^{k}\binom{k}{t}\frac{\left (-1\right )^{k-t+s}} {n^{2\beta +t+s}} \frac{\varGamma (2\beta + t + s)} {\varGamma (2\beta + t)} \sum _{m=1}^{n}\left [U^{\chi }\left (ST^{m\varepsilon }\right )\right ]_{ i,j} {}\\ & & \varPhi \left (\chi \left (r_{j}^{\left (n\right )}T^{n\varepsilon }\left (r_{ j}^{\left (n\right )}\right )^{-1}\right ),2\beta + t + s, \frac{m + 1} {n} \right ) {}\\ \end{array}$$