Variational and coupled-cluster calculations of the spectra of anharmonic oscillators

Energy Levels for Nonsymmetric Double-Well Potentials in Several Dimensions: Hill Determinant Approach

“…The difficulties originally encountered by Flessa [18,19] in connection with application of the Hill determinant approach were investigated by several workers [20][21][22][23][24][25], leading to some conclusions about the conditions of applicability of the approach. For example, Chaudhuri [20] treated anharmonic oscillators of the type (ax 2 ϩ bx 4 ϩ cx 6 ) and showed that, with a particular choice of convergence factor of the form exp(ϪͰx 4 ϩ ͱx 6 ), the algebraic of Hill determinant may…”

Iterative Bogoliubov transformations and anharmonic oscillators

“…For instance, it has been shown [3] that a Rayleigh-Schrodinger perturbation series for a quartic anharmonic oscillator diverges even for small values of the anharmonicity. Thus it is not surprising that many approximation techniques such as optimized variational methods [4,5], coupled-cluster calculations [5,6], WKB [7], and modified perturbational schemes [8,9] have been used for their study. An interesting consequence of such studies is the recognition that if one insists in taking a harmonic oscillator as a starting point for either a variational or a perturbative approximation to the problem, the best choice is usually not the harmonic oscillator that results from making the anharmonic terms equal to zero.…”

“…An interesting consequence of such studies is the recognition that if one insists in taking a harmonic oscillator as a starting point for either a variational or a perturbative approximation to the problem, the best choice is usually not the harmonic oscillator that results from making the anharmonic terms equal to zero. The selection of an alternative frequency (scaling) and an alternative position to center the harmonic oscillator (displace-ment) can either be done by demanding the correct behavior for the ground state [4,5] or may be made state dependent via the so-called principle of minimum sensitivity [10].…”

Airy function approach and Numerov method to study the anharmonic oscillator potentials V(x) = Ax2α + Bx2

“…Figures (4)(5)(6) show the variations of the functions F afa (E) and F nm (E) as a function of energy for the coefficients for A=1, B=-4.9497 and the power factor α=2, 3 and 4. These two functions meet the energy axis simultaneously and give practically the same energy eigenvalues.…”

“…Some methods have been established to determine with precision the energy spectra of many anharmonic oscillators. 1 They have been carried out using the Hill determinant, 2,3 the coupled cluster method, 4,5 the Bargmann representation, 6,7 the variational-perturbation expansion and other approaches. [8][9][10][11][12][13] Using the double exponential Sinc collocation method, Gaudreau et al 14 evaluated the energy eigenvalues of anharmonic oscillators.…”