Surprises in nonlinear perturbations: Case of a multiple well potential problem

Abstract

ABSTRACT:We study an atypical nonlinear perturbation problem on the lowest three states of a harmonic oscillator. The perturbation is such that the total potential is a single-well at low values of coupling strength, turns to a double-well in the intermediate coupling range and finally becomes a triple-well at high values of the coupling strength. Proceeding via logarithmic perturbation theory, we notice that the perturbation furnishes exact results for both the energy and wave function of the second excited state but shows divergence for the first excited one. However, the ground state wave function is found to be exactly summable to the correct result with a finite energy series. Effects of varying the coupling constant, and hence the number of wells, on energies and wave functions of the concerned states are noted. Our results show further that the perturbed ground state acquires a pre-exponential factor and, strangely, an exponent same as the second excited state. The origin of such an outcome is analyzed. We also highlight the reason behind the Klauder phenomenon that tells of increased sustenance of memory effects in wave functions, when compared with the Hamiltonian, in the limit of zero coupling strength.