A complete variational treatment is provided for a family of spiked-harmonic oscillator Hamiltonians H=−d2/dx2+Bx2+λ/xα(B>0,λ>0), for arbitrary α>0. A compact topological proof is presented that the set S={ψn} of known exact solutions for α=2 constitutes an orthonormal basis of the Hilbert space L2(0,∞). Closed-form expressions are derived for the matrix elements of H with respect to S. These analytical results, and the inclusion of a further free parameter, facilitate optimized variational estimation of the eigenvalues of H to high accuracy.

We evaluate the Casimir force between two parallel plates in Randall-Sundrum (RS) scenarios extended by $q$ compact dimensions. After giving exact expressions for one extra compact dimension (6D RS model), we generalize to an arbitrary number of compact dimensions. We present the complete calculation for both the two-brane scenario (RSI model) and the one-brane scenario (RSII model) using the method of summing over the modes. We investigate the effects of extra dimensions on the magnitude and sign of the force, and comment on limits for the size and number of the extra dimensions.

This paper presents a new class of fractional order Runge–Kutta (FORK) methods for numerically approximating the solution of fractional differential equations (FDEs). We construct explicit and implicit FORK methods for FDEs by using the Caputo generalized Taylor series formula. Due to the dependence of fractional derivatives on a fixed base point, in the proposed method, we had to modify the right-hand side of the given equation in all steps of the FORK methods. Some coefficients for explicit and implicit FORK schemes are presented. The convergence analysis of the proposed method is also discussed. Numerical experiments are presented to clarify the effectiveness and robustness of the method.

For the family of model soft-core Coulomb potentials represented by $V\left(r\right)=-\left[Z/{\left(r^q+{\beta }^q\right)}^{1/q}\right]$, with the parameters $Z>0,\beta >0,q\ge 1$, it is shown analytically that the potentials and eigenvalues, $E_{\nu \ell }$, are monotonic in each parameter. The potential envelope method is applied to obtain approximate analytic estimates in terms of the known exact spectra for pure power potentials. For the case $q=1$, the asymptotic iteration method is used to find exact analytic results for the eigenvalues $E_{\nu \ell }$ and corresponding wave functions, expressed in terms of $Z$ and $\beta$. A proof is presented establishing the general concavity of the scaled electron density near the nucleus resulting from the truncated potentials for all $q$. Based on an analysis of extensive numerical calculations, it is conjectured that the crossing between the pair of states $\left[\left(\nu ,\ell \right),\left({\nu }^{\prime },{\ell }^{\prime }\right)\right]$ is given by the conditions ${\nu }^{\prime }\ge \left(\nu +1\right)$ and ${\ell }^{\prime }\ge \left(\ell +3\right)$. The significance of these results for the interaction of intense laser field with an atom is pointed out. Differences in the observed level-crossing effects between the soft-core potentials and the hydrogen atom confined inside an impenetrable sphere are discussed.

We apply the Asymptotic Iteration Method to obtain the bound-state energy spectrum for the d-dimensional Klein-Gordon equation with scalar S(r) and vector potentials V(r). When S(r) and V(r) are both Coulombic, we obtain all the exact solutions; when the potentials are both of Kratzer type, we obtain all the exact solutions for S(r) = V(r); if S(r) > V(r) we obtain exact solutions under certain constraints on the potential parameters: in this case, a possible general solution is found in terms of a monic polynomial, whose coefficients form a set of elementary symmetric polynomials.