Uniqueness of solutions to Schr\"odinger equations on 2-step nilpotent Lie groups

Abstract

Let g=g_1+g_2, [g,g] =g_2, be a nilpotent Lie algebra of step 2, V_1,..., V_m a basis of g_1 and L=\sum_{j,k} a_{jk} V_j V_k be a left-invariant differential operator on G=exp (g), where the coefficients a_{jk} form a real, symmetric mxm-matrix. It is shown that if a solution w(t,x) to the Schr\"odinger equation \partial_t w(t,g)=i Lw(t,g), w(0,g)=f(g), satisfies a suitable Gaussian type estimate at time t= 0 and at some time t=T\ne 0, then w=0 . The proof is based on Hardy's uncertainty principle and explicit computations within Howe's oscillator semigroup. Our results extend work by Ben Said and Thangavelu in which the authors study the Schr\"odinger equation associated to the sub-Laplacian on the Heisenberg group.