Global solution of the cubic oscillator

TL;DR

This paper introduces a novel method for solving the Schrödinger equation globally to find eigenstates and eigenenergies of cubic anharmonic oscillators, including complex cases and scattering scenarios.

Contribution

It presents a new approach for obtaining global solutions of the Schrödinger equation for cubic oscillators, simplifying the calculation of eigenstates and energies.

Findings

Successfully determines eigenenergies and eigenfunctions for real and complex cubic oscillators.

Provides a simple method for solving the Schrödinger equation with boundary conditions.

Analyzes scattering phenomena in a real cubic potential.

Abstract

The eigenstates of a real or complex cubic anharmonic oscillator are investigated using original and alternative methods. The procedure consists of determining global solutions of the Schr\"odinger equation that comply with the pertinent boundary conditions and allows us to obtain, in a very simple way, the eigenenergies and eigenfunctions of the Hamiltonian. Scattering by a real cubic potential is investigated as a specific case.