Spectral data for finite volume hyperbolic surfaces at the bottom of the continuous spectrum (Q1331652)

The author considers the Laplace operator acting on some non-compact finite volume hyperbolic surfaces. The spectrum consists of a continuous part filling \([1/4,\infty)\) and a discrete set of eigenvalues of which finitely many are less or equal to \(1/4\). By definition, a cusp form is any \(L^ 2\) eigenfunction with zero Fourier coefficients. It is proved that for a generic hyperbolic Riemann surface with cusps and signature \((g,0,m)\), \(g + m \geq 3\), there are no cusp forms with eigenvalue \(\lambda = 1/4\). The singular part of the Green function at \(\lambda = 1/4\) is characterized explicitly in terms of the Eisenstein series and an orthonormal basis of the cusp forms with eigenvalue \(\lambda = 1/4\). A formula for the first variation of a resonance at the bottom of the continuous spectrum is given. Some examples are investigated.