Support theorem for the fundamental solution to the Schrödinger equation on certain compact symmetric spaces (Q624338)

The author considers the following Cauchy problem for the Schrödinger equation \[ \begin{cases} \sqrt{-1}\partial_t\psi+\Delta_M\psi=0,& t\in\mathbb R, \\ \psi(0,x)=\delta_o(x),\quad & x\in M, \end{cases} \] where \(M=U/K\) is a compact symmetric space with certain conditions on root multiplicities, on the weight lattice and the coroot lattice associated with the maximal torus \(A\subset M,\) \(\delta_o(x)\) is the Dirac's delta function with singularity at \(o=eK\in M,\) and \(\Delta_M\) is the Laplace-Beltrami operator on \(M\) with respect to the \(U\)-invariant metric. The fundamental solution to the problem is constructed using shift operators of Heckman and Opdam. Next, it is proved that the support of the fundamental solution becomes a lower dimensional subset at a rational time whereas its support and its singular support coincide with the whole symmetric space at an irrational time. Moreover, it is also shown that generalized Gauss sums appear in the expression of the fundamental solution.