On the energy spectra of one-dimensional anharmonic oscillators

Non-perturbative energy expressions for the generalized anharmonic oscillator

“…The above expressions are in exact agreement with the corresponding expression derived by Mathews et al [11]. Figures 3, 4 and 5 give the variation of E 1 (λ), E 2 (λ) and E 3 (λ) with v in the range 0.2 v 1, respectively.…”

Studies on some singular potentials in quantum mechanics

“…Therefore all table entries are to be taken as correct up to the place they are reported. The results used for comparison in Table I are, chronologically, (a) the linear variation method involving diagonalization of matrices of large order (800 × 800) 39, (b) the analytical formula based on the scaled oscillator approach 40, (c) the finite‐difference calculation 10, and (d) the asymptotic shooting method 41. As seen, the present result for all these states match exactly up to the 9th decimal place with the accurate results of Ref.…”

Critical temperature for Bose-Einstein condensation in quartic potentials

“…For the one dimensional quartic oscillator, eigen values can be calculated by minimizing the expectation of the Hamiltonian in the basis states of harmonic oscillator of appropriately chosen frequency [12]. Generalizing the result to three dimensional case, the eigen energy ǫ(n 1 , n 2 , n 3 ) = 1.389…”

“…For finite number of bosons, the zero point energy causes a change in the value of T c . For the 3D quartic potential, the ground state energy is [12] ǫ min = 2.41 λ .…”

“…In particular, the energy of the ground state is essential to estimate T c for finite number of particles. To estimate the correction to T c in the finite particle case, we calculate the ground state energy analytically [12]. The calculation is based on a method which optimizes the matrix elements of the quartic potential Hamiltonian in the harmonic oscillator basis.…”