Uniqueness of solutions to Schrödinger equations on 2-step nilpotent Lie groups

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 Schrödinger 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 Schrödinger equation associated to the sub-Laplacian on the Heisenberg group.

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