Nodal structure of bound-state wave functions for systems with quartic dispersion

Abstract

The nodal structure of bound-state wave functions for one-dimensional quantum systems with quartic energy-momentum dispersion and polynomial potentials is analysed by using the semiclassical approximation and variational approach. For energies of bound states, we derive the quantization condition, obtained by using the complex Wentzel method, where we take into account perturbative (up to the fourth order) and nonperturbative in the Planck constant corrections. The bound-state energies and wave functions for the harmonic and quartic potentials are compared with those found by applying the variational approach utilizing the universal Gaussian basis. It is shown that the classical oscillation theorem, valid for systems with quadratic energy-momentum dispersion, breaks down in the classically forbidden region where wave functions also have nodes, while it still remains valid in the classically allowed region. These results are confirmed in addition via the solutions of the exactly solvable problem of the fourth-order Schrodinger equation with a square well potential.