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

Abstract:

Let $\mathfrak {g}=\mathfrak {g}_1\oplus \mathfrak {g}_2,[\mathfrak {g},\mathfrak {g}] =\mathfrak {g}_2,$ a nilpotent Lie algebra of step 2, $V_1,\cdots , V_m$ a basis of $\mathfrak {g}_1$ and $L=\sum _{j,k}^{m}a_{jk}V_j V_k$ a left-invariant differential operator on $G=\mathrm {exp} (\mathfrak {g})$, where $(a_{jk})_{jk}\in M_n(\mathbb {R})$ is symmetric. It is shown that if a solution $w(t,x)$ to the Schrödinger equation $\partial _t w(t,g)=i Lw(t,g),$ $g\in G, t\in \mathbb {R}, w(0,g)=f(g)$, 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, on explicit computations within Howe’s oscillator semigroup and on methods developed by Fulvio Ricci and the second author. Our results extend work by Ben Saïd, Thangavelu and Dogga on the Schrödinger equation associated to the sub-Laplacian on Heisenberg type groups.