We give relations between main operators of quantum mechanics on one of most general classes of nilpotent Lie groups. Namely, we show relations between momentum and position operators as well as Euler and Coulomb potential operators on homogeneous groups. Homogeneous group analogues of some well-known inequalities such as Hardy's inequality, Heisenberg–Kennard type and Heisenberg–Pauli–Weyl type uncertainty inequalities, as well as Caffarelli–Kohn–Nirenberg inequality are derived, with best constants. The obtained relations yield new results already in the setting of both isotropic and anisotropic ℝn, and of the Heisenberg group. The proof demonstrates that the method of establishing equalities in sharper versions of such inequalities works well in both isotropic and anisotropic settings.
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机译：我们给出了最一般的幂等李群中的一类的量子力学主要算符之间的关系。即，我们显示了同质群上动量和位置算子以及欧拉和库仑势算子之间的关系。推导了一些众所周知的不等式的均质群类似物，它们具有最佳常数，这些不等式包括Hardy不等式，Heisenberg-Kennard型和Heisenberg-Pauli-Weyl型不确定性不等式以及Caffarelli-Kohn-Nirenberg不等式。所获得的关系已经在各向同性和各向异性ℝ n 以及Heisenberg群的设置中产生了新的结果。证明表明，在各向同性和各向异性环境中，在这种不等式的更尖锐版本中建立等式的方法都行之有效。
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