In this paper we consider uncertainty principles for solutions of certain partial differential equations on $H$ -type groups. We first prove that, on $H$ -type groups, the heat kernel is an average of Gaussians in the central variable, so that it does not satisfy a certain reformulation of Hardy’s uncertainty principle. We then prove the analogue of Hardy’s uncertainty principle for solutions of the Schrödinger equation with potential on $H$ -type groups. This extends the free case considered by Ben Saïd et al. [‘Uniqueness of solutions to Schrödinger equations on H-type groups’, J. Aust. Math. Soc. (3)95 (2013), 297–314] and by Ludwig and Müller [‘Uniqueness of solutions to Schrödinger equations on 2-step nilpotent Lie groups’, Proc. Amer. Math. Soc.142 (2014), 2101–2118].

中文翻译：

---

型群上薛定谔方程解的不确定性原理

在本文中，我们考虑了某些偏微分方程解的不确定性原理 $H$ 型组。我们首先证明，在 $H$ 型群，热核是中心变量中高斯的平均值，因此它不满足哈代不确定性原理的某种重新表述。然后我们证明了薛定谔方程解的哈代不确定性原理的类比 $H$ 型组。这扩展了 Ben Saïd 考虑的自由案例等。['H 型群薛定谔方程解的唯一性'，J.奥斯特。数学。社会党。(3)95(2013), 297–314] 和 Ludwig 和 Müller ['2-step nilpotent Lie groups 上薛定谔方程解的唯一性'，过程。阿米尔。数学。社会党。142（2014），2101-2118]。