Spectral Theory of Infinite-Area Hyperbolic Surfaces by David Borthwick

Author:David Borthwick
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

(9.67)

The functional relation S X (s)−1 = S X (1 − s) from Corollary 7.​14, which holds for S 0(s) as well, implies that

(9.68)

This also implies that S X (s) and S 0(s) are unitary for by (7.​36), so that

(9.69)

Note also that . By Lemma 8.​9, this implies that , where P is an orthogonal projection of rank . Hence

(9.70)

The relative scattering determinant will provide the crucial link between resonances and the regularized traces of the resolvent that we introduce in Chapter 10 The main goal of this section is to establish a Hadamard-type factorization formula for τ X (s), with a growth estimate.

In Theorem 9.2 we saw that the resonance counting function N X (r) is bounded O(r 2). By the Weierstrass factorization theory, this implies that the product

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