Scattering theory for the Laplacian on symmetric spaces of noncompact type and its application to a conjecture of Strichartz (Q1628401)

The scattering theory for the Laplacian on symmetric spaces \(X\) of noncompact type is developed in the framework of the Agmon-Hörmander space. The study is based on a detailed analysis of the Helgason Fourier transform and generalized spherical functions on symmetric spaces of noncompact type. Let \(n_X\) be the dimension of \(X\) and let \(\mathbf{H}_0\) denote the modified Laplacian on \(X\) whose spectrum coincides with the nonnegative real half-line. The first main theorem containing a scattering formula is stated as follows: Let \(\kappa\) be a positive real number and let \(u\in L^2_{\mathrm{loc}}(X)\) be an arbitrary solution to the homogeneous Helmholtz equation \((\mathbf{H}_0-\kappa^2)u=0\) that belongs to the Agmon-Hörmander space. Then there exists a unique pair of boundary values \(\varphi_\pm\) at infinity such that \[ u(x)\simeq(2\pi)^{-1/2}\mathcal{J}(x)^{-1/2}r(x)^{-(n_X-1)/2}\Big\{e^{+i\kappa r(x)}\varphi_+(\hat{x})+e^{-i\kappa r(x)}\varphi_-(\hat{x})\Big\}, \] where \(r(x)=d(x,o)\) is a distance function, \(\hat{x}\) is an \(X\)-analogue of \(\hat{x}=x/|x|\) in \(\mathbb{R}^n\), and \(\mathcal{J}(x)\) is a Jacobian in \(X\). In the second main theorem, the high-energy limit (\(\kappa\to\infty\)) and the low-energy limit (\(\kappa\downarrow 0\)) of the geometric scattering matrix \(\hat{S}_X(\kappa)\) defined by the mapping \(\varphi_-\mapsto\varphi_+\) are described. As an application of the constructed scattering theory, the author proved the Strichartz conjecture: An arbitrary family \(\{u_\kappa\}_{\kappa\in\mathbb{R}_+}\) of generalized eigenfunctions of the Laplacian \(\Delta_X\) in a suitable Banach space can be characterized as \(u_\kappa =\mathcal{P}_\kappa u\) by a spectral projection operator \(\mathcal{P}_\kappa\) and a unique \(L^2\)-function \(u\) on \(X\).