Scattering on \(p\)-adic graphs
In the paper the scattering problem being considered earlier for the real symmetric space is generalized for the case of non-Archimedean fields. The authors have studied the simplest \(p\)-adic symmetric space \(H^{(p)}= PGL(2,Q_p)/PGL(2,Z_p)\) realized as the Bruhat-Tits tree and its quotient spaces by some discrete subgroups realized as the \(p\)-adic Riemann surfaces. Here the real field case corresponds to \(p=\infty\). Earlier P. G. O. Freund (1991) developed a formal scattering theory on \(H^{(p)}\) only for zonal spherical functions (or \(s\)-waves) and obtained the corresponding \(S\)-matrix as a ratio of two \(p\)-adic Harish-Chandra \(c\)-functions. In the paper under consideration the authors study the general case of the scattering problem for automorphic functions on \(H^{(p)}\), i.e. the scattering problem for the Laplace operator on the \(p\)-adic Riemann surfaces in the frame of spectral theory. First of all they construct the orthogonal decomposition of the Laplace operator on the Bruhat-Tits tree and the \(p\)-adic Riemann surface and as result of that they reduce the scattering problem on these spaces to the set of one-dimensional scattering problems for partial channels. This allows to develop for all channels both the stationary scattering theory and the nonstationary Lax-Phillips approach in a closed form. The \(S\)-matrix singularities are proved to coincide with zeros of the zeta-function. In the case of the Bruhat-Tits tree the resonance structure appears only in the \(s\)-wave channel. As an illustrative example of the theory with nontrivial discrete subgroup the authors consider the scattering on the Riemann surface of genus one. In the 1-loop case they draw the scattering amplitudes also from asymptotics of the Eisenstein series and compare them with those obtained in the frame of the Lax-Phillips approach.
scientific article; zbMATH DE number 1135983