Narrow Results

The SU(1,1) generators are considered in order to investigate the uncertainty relation in the Barut–Girardello coherent state for the time-dependent singular oscillator. The corresponding uncertainty product is obtained by evaluating the expectation values of the canonical variables and their squares.

In this paper, asymptotic behavior of the uncertainty product of the Gauss–Weierstrass wavelet is investigated. It is shown that the uncertainty product is bounded from above, a feature that distinguishes the Gauss–Weierstrass wavelet from other wavelet families.

We study a localization of functions defined on Vilenkin groups. To measure the localization, we introduce two uncertainty products $U{P}_{\lambda }$ and $U{P}_G$ that are similar to the Heisenberg uncertainty product. $U{P}_{\lambda }$ and $U{P}_G$ differ from each other by the metric used for the Vilenkin group $G$. We discuss analogs of a quantitative uncertainty principle. Representations for $U{P}_{\lambda }$ and $U{P}_G$ in terms of Walsh and Haar basis are given.