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Mathematics, Spectral Theory, math.SP,
Abstract:
We study the isoresonance problem on non-compact surfaces of finite area that
are hyperbolic outside a compact set. Inverse resonance problems correspond to
inverse spectral problems in the non-compact setting. We consider a conformal
class of surfaces with hyperbolic cusps where the deformation takes place
inside a fixed compact set. Inside this compactly supported conformal class we
consider isoresonant metrics, i.e. metrics for which the set of resonances is
the same, including multiplicities. We prove that sets of isoresonant metrics
inside the conformal class are sequentially compact. We use relative
determinants, splitting formulae for determinants and the result of B. Osgood,
R. Phillips and P. Sarnak about compactness of sets of isospectral metrics on
closed surfaces.
In the second part, we study the relative determinant of the Laplace operator
on a hyperbolic surface as function on the moduli space. We consider the moduli
space of hyperbolic surfaces of fixed genus and fixed number of cusps. We
consider the relative determinant of the Laplace operator and a model operator
defined on the cusps. We prove that the relative determinant tends to zero as
one approaches the boundary of the moduli space.
