Lax-Phillips Scattering for Atomorphic Functions Based on the Eisenstein Transform |
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Authors: | Yoichi Uetake |
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Institution: | (1) Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Umultowska 87, PL-61-614 Poznań, Poland |
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Abstract: | We construct a Lax-Phillips scattering system on the arithmetic quotient space of the Poincaré upper half-plane by the full
modular group, based on the Eisenstein transform. We identify incoming and outgoing subspaces in the ambient space of all
functions with finite energy-form for the non-Euclidean wave equation. The use of the Eisenstein transform along with some
properties of the Eisenstein series of two variables enables one to work only on the space corresponding to the continuous
spectrum of the Laplace-Beltrami operator. It is shown that the scattering matrix is the complex function appearing in the
the functional equation of the Eisenstein series of two variables. We obtain a compression operator constructed from the Laplace-Beltrami
operator, whose spectrum consists of eigenvalues that coincide, counted with multiplicities, with the non-trivial zeros of
the Riemann zeta-function. For this purpose we construct and use a scattering model on the one-dimensional Euclidean space.
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Keywords: | 11F03 11F72 47A11 47A40 |
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