In this paper we study, both analytically and numerically, questions involving the distribution of eigenvalues of Maass forms on the moonshine groups $Gamma_0(N)^+$, where $N>1$ is a square-free integer. After we prove that $Gamma_0(N)^+$ has one cus
p, we compute the constant term of the associated non-holomorphic Eisenstein series. We then derive an average Weyls law for the distribution of eigenvalues of Maass forms, from which we prove the classical Weyls law as a special case. The groups corresponding to $N=5$ and $N=6$ have the same signature; however, our analysis shows that, asymptotically, there are infinitely more cusp forms for $Gamma_0(5)^+$ than for $Gamma_0(6)^+$. We view this result as being consistent with the Phillips-Sarnak philosophy since we have shown, unconditionally, the existence of two groups which have different Weyls laws. In addition, we employ Hejhals algorithm, together with recently developed refinements from [31], and numerically determine the first $3557$ of $Gamma_0(5)^+$ and the first $12474$ eigenvalues of $Gamma_0(6)^+$. With this information, we empirically verify some conjectured distributional properties of the eigenvalues.
We explore an algorithm which systematically finds all discrete eigenvalues of an analytic eigenvalue problem. The algorithm is more simple and elementary as could be expected before. It consists of Hejhals identity, linearisation, and Turing bounds.
Using the algorithm, we compute more than one hundredsixty thousand consecutive eigenvalues of the Laplacian on the modular surface, and investigate the asymptotic and statistic properties of the fluctuations in the Weyl remainder. We summarize the findings in two conjectures. One is on the maximum size of the Weyl remainder, and the other is on the distribution of a suitably scaled version of the Weyl remainder.
Asking for the optimal protocol of an external control parameter that minimizes the mean work required to drive a nano-scale system from one equilibrium state to another in finite time, Schmiedl and Seifert ({it Phys. Rev. Lett.} {bf 98}, 108301 (200
7)) found the Euler-Lagrange equation to be a non-local integro-differential equation of correlation functions. For two linear examples, we show how this integro-differential equation can be solved analytically. For non-linear physical systems we show how the optimal protocol can be found numerically and demonstrate that there may exist several distinct optimal protocols simultaneously, and we present optimal protocols that have one, two, and three jumps, respectively.
