Perturbative, asymptotic and Pade-approximant solutions for harmonic and inverted oscillators in a box

Spherical confinement of coulombic systems inside an impenetrable box: H atom and the Hulthén potential

“…A Hartree-Fock self-consistent field solution [13] with Slater-type orbitals and cut-off functions has been proposed. Approximate analytical formulas for CHA eigenvalues were derived using Vawter's cothz method, [14] joint perturbation method, and Pad e approximation, [15,16] Wentzel-Kramers-Brillouin (WKB) method. [17] Some other attempts are: a combined hypervirial theorem and perturbation theory, [18] a variational boundary perturbation method with appropriate cut-off function [19,20] along with its variants, [21,22] by extending a power-series solution, [23] originally proposed for free quantum systems, to confined case, [24] selfconsistent solution [25] of relevant Kohn-Sham equation within the broad domain of density functional theory, variational perturbation theory, [26] variational method in conjunction with super-symmetric quantum mechanics, [27,28] Rayleigh-Schr€ odinger perturbation theory, [29] Lie algebraic treatment, [29] Lagrange-mesh method, [30] searching the zeros of hypergeometric function, [31] asymptotic iteration method, [32] and so forth.…”

Quantum confinement in 1D systems through an imaginary-time evolution method

“…Eigenvalues were numerically presented as roots of a polynomial as well [22]. Padé approximants constructed as interpolations between perturbative and asymptotic solutions were proposed in [23]. Diagonal hypervirial relations [24] as well as hypervirial perturbative method [25] were proposed.…”

Confinement in 3D polynomial oscillators through a generalized pseudospectral method

“…For example, the effect of finite boundaries on a 1D CHO inside a potential enclosure was studied by means of WKB method [10]. Perturbative, asymptotic and Padé approximant solutions for boxed-in harmonic and inverted oscillators were reported [11]. Eigenfunctions, eigenvalues in such 1D potentials were also obtained by means of a hypervirial method [12,13].…”