Spectral geometry and scattering theory for certain complete surfaces of finite volume. (English) Zbl 0772.58063

The author studies complete surfaces \((M,g)\) of finite area whose Gaussian curvature equals \(-1\) in the complement of some compact subset of \(M\). The spectrum of the Laplacian consists of a sequence of (nonnegative) eigenvalues and an absolutely continuous spectrum. The stationary approach to scattering theory gives rise to a scattering matrix \(C(s)\) i.e. a meromorphic matrix valued function of \(s \in C\). The poles and zeros of \(\phi(s)=C(s)\) are called resonances. \(\sigma(M)\) is the union of: (a) the resonances; (b) the set of all complex \(s_ j\) such that \(\lambda_ j=s_ j(1-s_ j)\) is an eigenvalue of \(\Delta\); (c) \(\{1/2\}\). The points \(\eta\) of \(\sigma(M)\) occur with a certain multiplicity \(m(\eta)\). It is shown that \(\phi(s)\) is a meromorphic function of order 2 and that moreover an analogue of the Weyl’s formula holds \[ \sum_{\eta\in\sigma(M); |\eta|\leq T}m(\eta) \sim {\text{Area}(M)\over 2\pi}T^ 2. \] Using a variant of Selberg’s trace formula the author can prove that \(\sigma(M)\) determines also the Euler characteristic \(\chi(M)\) and the number \(m\) of ends. The resonance zeta function \(\zeta_ B(s)\) is defined by \[ \zeta_ B(s)=\sum_{\eta \in\sigma(M); \eta\neq 1}(1-\eta)^{-s} \] while \(\zeta_ \Delta\) is the spectral zeta function of the Laplacian. Correspondingly we get two regularized determinants \[ det'(\Delta)=e^{-\zeta'_ \Delta(0)}\text{ and }det'B_ 1=e^{-\zeta'_ B(0)} \] which are related by \[ det'\Delta=exp\left({\text{Area}(M)\over 8\pi}-{3\pi\gamma\over 2}m\right)det'B_ 1 \] where \(\gamma\) is Euler’s constant. For \(\text{Re}(z) > 1\) the author introduces the regularized determinant \(\text{det}(B_ 1+(z-1))\). When \(M\) is hyperbolic this determinant is described in terms of \(\phi(s)\) and the Selberg zeta function.


58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35P25 Scattering theory for PDEs
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
Full Text: DOI EuDML


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