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

2012/07/19 by Jean Ludwig, Ludwig, Jean, Detlef Müller +1 · 1 citation
Mathematics · #22E25 #22E30 #35B05 #43A80 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Spectral Theory in Mathematical Physics #math.AP #math.CA #msc:22E25 #msc:22E30 #msc:35B05 #msc:43A80

paper · pdf · doi:10.48550/arxiv.1207.4652

15 pages

arxiv created 2012/07/19 · openalex publication_date 2012/07/19 · arxiv updated 2012/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let g=g₁+g₂, [g,g] =g₂, be a nilpotent Lie algebra of step 2, V₁,..., V_m a basis of g₁ and L=∑_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 ∂_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≠ 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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