A quantum-mechanically solvable nonpolynomial Lagrangian with velocity-dependent interaction

Abstract

The quantum-mechanical problem of the nonlinear oscillator with the Lagrangian L= ½[x^.2-k₀x ²)/(l-λx²)] is solved exactly and the energy levels and eigenfunctions are obtained completely. This model (whenk 0=0) is the zero-space-dimensional isoscalar analogue of the nonlinear SU₂⊗SU₂ chirally invariant Lagrangian in the Gasiorowicz-Geffen co-ordinates and may also be considered as a modified version of the anharmonic-oscillator and Lee-Zumino models. The bound-state energy levels are found to have a linear dependence on the coupling parameter, in sharp contrast to the case of the familiar oscillator With quartic anharmonicity where the energy, as a function of λ, has complicated singularities at λ = 0. We investigate how far certain standard approximation procedures reproduce the exact results. The Bohr-Sommerfeld quantization procedure is found to reproduce the form of the boundstate energy levels correctly. Interestingly a perturbation-theoretic treatment also reproduces the correct results at least up to the order (λ²) to which we have carried our calculations.