On the resolvent of the Laplacian on functions for degenerating surfaces of finite geometry
Michael Schulze
Abstract
We consider families of degenerating hyperbolic surfaces. The surfaces are geometrically finite of fixed topological type. Let Z(s) be the Selberg Zeta function of a surface, and let Zd(s) be the contribution of the pinched geodesics to the Zeta function. Extending a result of Hejhal and Wolpert, we prove that the quotient of these two terms converges to the Zeta function of the limit surface for all arguments s with re(s)>1/2. The technique is an examination of resolvent of the Laplacian, which is composed from that for elementary surfaces via meromorphic Fredholm theory. The resolvent is shown to converge on the complement of the essential spectrum of the limit surface. We also use this property to define approximate Eisenstein functions and scattering matrices.
Create a lesson
Related papers
A note on the BRST-BV Formalism Explanation of Deformation Quantization and Algebraic Index Theorem
Lihao Zhao
Biharmonic submanifolds in nonpositive space forms
Yu Fu, Min-chun Hong, Dan Yang
Conjugate radius and parabolicity along the Ricci flow
Chandan Kumar Mondal
Pure Spinor Factorization of Belinski-Zakharov Solitons and Regular Gravitational Instantons
Steffen Aksteiner
Flexibility of Laplace-Beltrami eigenfunctions under metric variation
Andrei Agrachev, Nicola Da Ponte, Antonio Lerario et al.
C1-rigidity of Hp,q-convex cocompact representations of Gromov hyperbolic groups
Nikolay Bogachev