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A formal analytic solution of the time-dependent Schrödinger equation is obtained for cutoff Gaussian wavepackets incident on finite range potentials of arbitrary shape. We derive closed analytical expressions for transmitted wavepackets with a small cutoff and the solutions are applied to investigate the dynamics of wavepacket tunneling. We showed that the analytic solutions satisfy a useful rescaling property that was applied to study and characterize the delay time in delta potential barriers.

Type 2 diabetes (T2D) is a major health problem worldwide. In this condition, the organism can produce insulin but becomes resistant to it; thus the insulin is ineffective. High blood glucose levels are a result of insulin resistance and insulin deficiency; they produce diabetes-associated complications such as kidney failure, blindness, cardiovascular disease, and lower-limb amputation. In Guatemala, there were over 752.700 cases of the disease in 2017 with prevalence of 8.4 (IDF, 2017). The use of plants for medicinal purposes has been practiced in the country since pre-Hispanic times. Among the Cakchiquels, the aerial parts of Hamelia patens Jacq., Neurolaena lobata (L.) R.Br. ex Cass., and Solanum americanum Mill. and the cortex of Croton guatemalensis Lotsy are used to treat type 2 diabetes. The aim of the present study was to confirm the hypoglycemic effect of the plants under normal conditions and under maltose and sucrose tolerance tests, as well as to test the activity of the plant extracts in vitro against alpha-glucosidases types I and II. In agreement with the traditional usage of the plants, in normal conditions without a sugar load, the extracts produced a statistically significant hypoglycemic effect similar to the control drug glibenclamide. When the sugar load was maltose, only Croton and Solanum produced a statistically significant (p < 0.05) hypoglycemic effect compared to the control drug, but when the sugar was sucrose, Croton and Hamelia produced a statistically significant effect (p < 0.05) beginning at 30 min compared to the control group, while Solanum did so at 60 min and Neurolaena at 90 min. In vitro assays showed that the extracts inhibited yeast alpha-glucosidases but not the rat intestinal ones. Of the tested plants, Croton exert an effect both under sugars' tests and under a normal tolerance test; these results suggest the potential use of this plant. The results presented here provided evidence based on the use of these plants as hypoglycemic agents in the treatment of type 2 diabetes.

We explore the dynamics of two-dimensional massless Dirac-fermions within a quantum shutter approach, which involves the time-evolution of an initial cut-off plane wave. We show that the probability density is governed by an interplay between diffraction in time and Zitterbewegung phenomena, typical of relativistic quantum shutter systems with nonzero mass. The time-diffraction appears as an oscillatory pattern in the probability density, similar to the effect predicted by Moshinsky in 1952 [Phys. Rev. 88, 625] for Schrödinger free matter-waves. The Zitterbewegung manifests itself as high-frequency oscillations embedded in the time-diffraction profile. We found that these two transient effects are induced by the transverse momentum component of the incident wave, ky, that acts as an effective mass of the system. Furthermore, this effective mass can be manipulated by tuning the incidence angle of the initial quantum state, which allows to control the frequencies of the transients. In particular, we demonstrate that near a normal incidence condition, the Zitterbewegung appears as a series of quantum beats in the probability density, with a beating frequency 2kyvF , where vF is the Fermi velocity.

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