We study a general double Dirac delta potential to show that this is the simplest yet versatile solvable potential to introduce double wells, avoided crossings, resonances and perfect transmission (T = 1). Perfect transmission energies turn out to be the critical property of symmetric and antisymmetric cases wherein these discrete energies are found to correspond to the eigenvalues of Dirac delta potential placed symmetrically between two rigid walls. For well(s) or barrier(s), perfect transmission [or zero reflectivity, R(E)] at energy E = 0 is non-intuitive. However, earlier this has been found and called "threshold anomaly". Here we show that it is a critical phenomenon and we can have 0 ≤ R(0) < 1 when the parameters of the double delta potential satisfy an interesting condition. We also invoke zero-energy and zero curvature eigenstate (ψ(x) = Ax + B) of delta well between two symmetric rigid walls for R(0) = 0. We resolve that the resonant energies and the perfect transmission energies are different and they arise differently. * Electronic address: 1:zahmed@barc.gov.in, 2: Sachinv@barc.gov.in, 3: mayank.

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We present one dimensional potentials V (x) = V 0 [e 2|x|/a − 1] as solvable models of a well (V 0 > 0) and a barrier (V 0 < 0). Apart from being new addition to solvable models, these models are instructive for finding bound and scattering states from the analytic solutions of Schrödinger equation. The exact analytic (semi-classical and quantal) forms for bound states of the well and reflection/transmission (R/T ) coefficients for the barrier have been derived. Interestingly, the crossover energy E c where R(E c ) = 1/2 = T (E c ) may occur below/above or at the barrier-top.A connection between poles of these coefficients and bound state eigenvalues of the well has also been demonstrated. * Electronic address: 1:zahmed@barc.gov.in, 2

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In this paper, we study two fractional models in the Caputo–Fabrizio sense and Atangana–Baleanu sense, in which the effects of malaria infection on mosquito biting behavior and attractiveness of humans are considered. Using Lyapunov theory, we prove the global asymptotic stability of the unique endemic equilibrium of the integer-order model, and the fractional models, whenever the basic reproduction number [Formula: see text] is greater than one. By using fixed point theory, we prove existence, and conditions of the uniqueness of solutions, as well as the stability and convergence of numerical schemes. Numerical simulations for both models, using fractional Euler method and Adams–Bashforth method, respectively, are provided to confirm the effectiveness of used approximation methods for different values of the fractional-order [Formula: see text].

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