Continuity of small eigenfunctions on degenerating Riemann surfaces with hyperbolic cusps (Q1908580)

The main purpose of this paper is to prove the following theorem. Theorem. For any finite volume hyperbolic Riemann surface \(M\), let \(\{\phi_{M,n}\}\) denote the set of eigenfunctions on \(M\) with eigenvalue \(\lambda_{M,n} < 1/4\). Let \(M_\ell\) be a degenerating family of connected hyperbolic Riemann surfaces of finite volume which converges to the limit surface \(M_0\). Let \(x\) and \(y\) be points on \(M_\ell\). a) Let \(f\) be any function on \(\mathbb{R}^+\) such that for some \(t > 0\), the Laplace transform \({\mathcal L} (f)(z)\) is in \(L^1\) on the vertical line \(z = t + i\sigma\). Let \(x\) and \(y\) be bounded away from the developing cusps. Then for any \(T < 1/4\), we have \[ \lim_{\ell \to 0}\biggl(\sum_{\lambda_{M_\ell,n} \leq T} f(T - \lambda_{M_\ell,n}) \phi_{M_\ell, n} (x) \phi_{M_\ell,n} (y)\biggr) = \sum_{\lambda_{M_0,n} \leq T} f(T - \lambda_{M_0,n}) \phi_{M_0,n} (x) \phi_{M_0,n} (y). \] Further, the convergence is uniform on compact sets of \(M_0 \times M_0\). b) Let \(0 \leq T < 1/4\) be a number which is not equal to an eigenvalue on \(M_0\). If \(x\) is bounded away from the developing cusps, then \[ \lim_{\ell \to 0}\biggl(\sum_{\lambda_{M_\ell,n\leq T}} \phi_{M_\ell,n} (x)^2\biggr) = \sum_{\lambda_{M_0,n \leq T}} \phi_{M_0,n} (x)^2. \] In particular, if the eigenspace associated to the eigenvalue \(\lambda_{M_0,n}\) is one dimensional, then \[ \lim_{\ell \to 0} \phi_{M_\ell,n} (x) = \phi_{M_0,n} (x). \] The convergence is uniform on compact sets of \(M_0\).