Tight upper and lower bounds for energy eigenvalues of the Schrödinger equation

Abstract

A method is presented for the calculation of tight upper and lower bounds for the energy eigenvalues of the Schrodinger equation. The method is based on a rational functional approximation for the series expansion of the solution of the Riccati equation for the logarithmic derivative of the wave function. Specific applications for one-dimensional anharmonic oscillators and for the Yukawa potential are given, and the present results are compared with those obtainable by other procedures.

Rayleigh-Ritz variational method with suitable asymptotic behaviour

“…A reasonable estimate of the rate of convergence is the logarithmic error L N = log E ( ) − E (RPM) where E ( ) is the eigenvalue calculated by any of the methods described in this paper and E (RPM) is a very accurate result obtained by means of the Riccati-Padé method [4,5]. Figure 2 shows that the rate of convergence of the RRM for the potential (8) is also exponential but in this case it decreases according to S 3 ( ) > S 2 ( ) > S 1 ( ).…”

Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators

“…Typically the accuracy obtained was about 8 significant figures. Some time later, in [32], the excited state energies have been estimated for the pure quartic and sextic AO (i.e. β = 0, m = 2 and 3), eleven significant figures were obtained on the estimate of E 0 for the quartic AO.…”

“…In general it is sufficient to choose one of the three values ω = 1, 0, −1 according to whether the function to be determined goes to ±∞, a constant or 0 when z → ∞. Eventually, considering two successive values of ω gives upper and lower bounds on the eigenvalues [32,10,33]. The advantage of the Padé method is that it is well adapted to reproduce, from its Taylor expansion, the analytic structure of a meromorphic function especially if it has poles.…”

Accurate calculation of the eigenvalues of a new simple class of superpotentials in SUSY quantum mechanics

“…However, those variational results are not sufficiently accurate and, as argued above, we are not aware of any proof of convergence. In order to obtain more accurate results we resort to the RPM [2][3][4]. To this end we consider the modified logarithmic derivative of the wavefunction ψ…”

“…First we improve the variational method proposed by Marques et al [1] and, second, we show that the Riccati-Padé method (RPM) [2][3][4] provides highly accurate results for the eigenvalues of those partner Hamiltonian operators. In addition, we show that exactly the same Rayleigh-Ritz variational method applied to the SUSY partner Hamiltonians is also suitable for the calculation of the eigenvalues of the quartic anharmonic oscillator.…”