Vibrational-rotational analysis of supersingular plus quadraticA/r 4+r 2 potential

Abstract

Most work on supersingular potentials has focused on the study of the ground state. In this paper, a global analysis of the ground and excited states for the successive values of the orbital angular momentum of the supersingular plus quadratic potential is carried out, making use of centrifugal plus quadratic potential eigenfunction bases. First, the radially nodeless states are variationally analyzed for each value of the orbital angular momentum using the corresponding functions of the bases; the output includes the centrifugal and frequency parameters of the auxiliary potentials and their eigenfunction bases. In the second stage, these bases are used to construct the matrix representation of the Hamiltonian of the system, and from its diagonalization the energy eigenvalues and eigenvectors of the successive states are obtained. The systematics of the accuracy and convergence of the overall results are discussed with emphasis on the dependence on the intensity of the supersingular part of the potential and on the orbital angular momentum.

A basis for variational calculations inddimensions

“…[41] Ref. [38] E U 0.001 3.004 075 (30) 3.004 074 (30) 3.004 047 (5) 3.004 04 3.004 022 (14) 0.01 3.039 409 (30) 3.039 244 (30) 3.037 474 (5) 3.037 43 3.036 744 (15) 0.1 3.302 485 (30) 3.296 024 (30) 3.269 700 (5) 3.269 28 3.266 874 (18) 1 4.329 449 (30) 4.323 263 (30) 4.318 963 (5) 4.318 54 4.317 311 (16) 10 7.735 136 (30) 7.735 114 (30) 7.735 596 (5) 7.735 32 7.735 111 (8) 100 17.541 890 (30) 17.541 890 (30) 17.542 040 (5) 17.541 92 17.541 890 (11) 1000 44.955 485 (30) 44.955 485 (30) 44.955 517 (5) 44.955 49 44.955 485 (4) , for a wide range of values of A = l(l + 1) and λ , using the present work E U and the bounds E U a obtained by Aguilera-Navarro et al [36] (see also [48] and [28]).…”

“…Table 4. A comparison between the upper bounds for the Hamiltonian H = − d 2 dr 2 + r 2 + A r 2 + λ r 4 , for a wide range of values of A = l(l + 1) and λ , using the present work E U and the bounds E U a obtained by Aguilera-Navarro et al [36] (see also [48] and [28] Table 7. Upper bounds for the Hamiltonian H = − d 2 dr 2 − γr 2 + r 4 with different values of γ. E 1 (V ) and E 3 (V ) represent the values obtained from the variational method discussed by Broges et al, and E U 1 and E U 3 are from the present work (with a 10 × 10 -matrix).…”

“…The ground-state of (4.3) with α = 4 then follows by diagonalization of H in the nonorthogonal basis. This particular study was then extended [36] to provide a global analysis of the ground and excited states for the successive values of the orbital angular momentum of the super-singular plus quadratic potential r 2 + λ/r 4 . Another variational study of the ground state of (4.2) was introduced by Hall et al [38] where three parameters trial functions ψ(r) = r p+ǫ exp(−βr q ), p = (α − 1)/2 were used to approximate upper bounds of the ground-state of (4.3)…”

Accurate solutions for the spiked oscillators

“…The singular or spiked oscillator is defined by the family of quantum Hamiltonians where α and λ take positive values only and the domain of x is [0,∞). This problem has received enough attention since the middle 70s 1–42 because of its applications in physical chemistry and nuclear physics 1, 3, 38–42 and mainly for its interesting mathematical behavior. As Aguilera‐Navarro and Guardiola pointed out 9, none of the two terms on the potential dominate for the extreme values of λ.…”

“…The number of works on this topic has been steadily increasing 1–42. A variety of methods have been used to solve this problem, and few of them are as follows: perturbation theory, the linear variational method, Rayleigh‐Ritz variational method, and numerical approaches.…”