251. Heisenberg’s uncertainty principle associated with the Caputo fractional derivative
- Author
-
Pan Lian
- Subjects
Sequence ,Pure mathematics ,Commutator ,Uncertainty principle ,010102 general mathematics ,Zero (complex analysis) ,Statistical and Nonlinear Physics ,Space (mathematics) ,01 natural sciences ,Fractional calculus ,0103 physical sciences ,Order (group theory) ,010307 mathematical physics ,0101 mathematics ,Mathematical Physics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
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 [Dq,zq]. We prove that the sequence of eigenvalues of [Dq,zq] 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.
- Published
- 2021