This paper revisits the generalized pseudospectral (GPS) method on the calculation of various radial expectation values of atomic systems, especially on the spatially confined hydrogen atom and harmonic oscillator. As one of the collocation methods based on global functions, the powerfulness and robustness of the GPS method has been well established in solving the radial Schrödinger equation with high accuracy.However, in our recent work, it was found that the previous calculations based on the GPS method for the radial expectation values of confined systems show significant discrepancies with other theoretical methods. In this work we have tackled such a problem by tracing its source to the GPS method and found that the method itself may not be able to obtain the system wave function at the origin. Combined with an extrapolation method developed here, the GPS method can fully reproduce the radial quantities obtained by other theoretical methods, but with more flexibility, efficiency, and accuracy. We apply the GPS-extrapolation method to investigate the relatievistic fine structure and hyperfine splitting of confined hydrogen atom in s-wave states where the zero-point wave function dominates. Good agreement with previous predictions is obtained for confined hydrogen in low-lying states, and benchmark results are obtained for high-lying excited states. The perturbation treatment of the fine and hyperfine interactions is validated in the confining environment. K E Y W O R D Sconfined atom, fine structure, generalized pseudospectral method, hyperfine splitting, radial expectation value | INTRODUCTIONIn the past few years, the generalized pseudospectral (GPS) method, which is also known as the semispectral method or collocation method, has been successfully applied to investigate the structural properties of atomic and molecular systems, [1][2][3][4][5] the quantum scattering processes in nuclear and atomic physics, [6,7] and the laser-atom interactions. [8][9][10] Its usefulness has been continuously revealed in recent years in accurately and efficiently solving both the time-independent and time-dependent Schrödinger and Dirac equations. Being a type of discrete variable representation (DVR) method, the GPS method shows its special superiority over other generalizations of DVR, such as the finite difference and finite element methods. For example, the Numerov and Runge-Kutta methods, as in the latter case, are local approaches to the unknown function by a sequence of overlapping low-order polynomials in a small subset of user-defined grid points. [11,12] The GPS method and its variants in different forms are generally global approaches to the unknown function using global basis functions with a high degree, for example, the trigonometric

The electric multipole polarizabilities of one-electron atoms embedded in weakly coupled Debye plasmas are calculated in the non-relativistic framework. The static dipole, quadrupole, octopole, and hexadecapole polarizabilities for hydrogen atoms in both ground and excited states at a variety of Debye screening parameters are calculated in high precision based on the sum-over-states method, where the system bound and continuum states are produced by employing the generalized pseudospectral method. It is shown that the contribution of bound states to the polarizability decreases with increasing the plasma screening strength, whereas the contribution of continuum states is enhanced. At very small screening parameters where the plasma environment starts to take effect, it is found that the 2l-pole polarizability for s-wave states with principle quantum number n≥l+1 has an abrupt change from its non-screening value to infinity. We attribute such a phenomenon to the sudden non-degeneracy of different angular momentum states in the n shell. With continuously increasing the screening strength, the polarizabilities for n≥l+1 states decrease to certain values and, eventually, they approach to infinity at the critical screening parameter. For states with n≤l, the 2l-pole polarizabilities show regular enhancement from the non-screening value to infinity. The present results are compared with other theoretical calculations available in the literature and it is shown that our work has established by now the most accurate predictions of multipole oscillator strengths and polarizabilities for one-electron atoms in Debye plasmas.

Binding energies, oscillator strengths, and polarizabilities of hydrogen-like atoms embedded in dense quantum plasmas are calculated in high precision by employing the generalized pseudospectral method. Benchmark predictions of these quantities are obtained for both the ground and excited states including high-order transitions. The critical behaviour of system eigenenergies, radial mean values, oscillator strengths, and multipole polarizabilities for bound states when they approach to the corresponding continuum limit as increasing the plasma screening strength are investigated in detail. It is shown that near the critical screening parameters the quantities in s-wave states show distinct behaviour from those in non-s states and such phenomenon is attributed to the divergent asymptotic form of wave functions caused by the centrifugal potential of orbital angular momentum. Based on the power laws of energies and radial mean values extracted from numerical calculations and the approximate formulae for dipole polarizabilities, the critical behaviour of the 2 k -pole polarizability of s-wave states is explicitly derived and numerically validated to follow a power law with the exponent −2(k + 1).

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