On the spectrum of the Hecke groups (Q1058015)

Let \({\mathfrak G}(\mu)\) (\(\mu\geq 1)\) denote the Hecke group generated by the transformations \(z\mapsto -1/z\), \(z\mapsto z+2\mu\) acting on the upper half-plane \({\mathbb{H}}\). Let \(\delta\) (\(\mu)\) be the Hausdorff dimension of the limit set of \({\mathfrak G}(\mu)\), and let \(\lambda\) (\(\mu)\) be the lowest eigenvalue of the operator -\(\Delta\) acting on \(L^ 2({\mathfrak G}(\mu)\setminus {\mathbb{H}})\), where \(\Delta\) denotes the Laplacian for the hyperbolic metric on \({\mathbb{H}}.\) Then it is known that \(\lambda =\delta (1-\delta)\) and that \(\lambda\) is strictly increasing, beginning at 0 when \(\mu =1\) and tending to 1/4 as \(\mu\) \(\to \infty\). The main result of the present work is: The function \(\lambda\) : ]1,\(\infty [\to {\mathbb{R}}\) is analytic and downwards convex. The proof is based on the theory of partial differential equations.