In this paper, we establish the Heisenberg’s uncertainty inequality associated with the Caputo derivative of order q ∈ (1, ∞) in the generalized Bargmann–Fock space. We also determine exactly when the equality occurs in the uncertainty inequality. It is done by estimating the growth of the eigenvalues of the commutator $[{\mathcal{D}}^q,z^q]$. We prove that the sequence of eigenvalues of $[{\mathcal{D}}^q,z^q]$ tends to ∞ when the fractional order q belongs to (1, ∞). However, the sequence converges to zero when q belongs to (0, 1), which shows different behavior. Hence, only a weak uncertainty inequality is obtained for the latter case.

中文翻译：

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海森堡的不确定性原理与Caputo分数导数相关

在本文中，我们建立与命令的卡普托衍生物相关联的海森堡的不确定性不等式q ∈（1，∞在广义巴格曼-福克空间）。我们还确定不确定性不平等中何时出现平等。通过估算换向器特征值的增长来完成$[{\mathcal{d}}^q，{ž}^q]$。我们证明了特征值的序列$[{\mathcal{d}}^q，{ž}^q]$趋于∞时分数阶q属于（1，∞）。但是，当q属于（0，1）时，序列收敛为零，这表明行为不同。因此，对于后一种情况，仅获得了弱的不确定性不等式。