From quartic anharmonic oscillator to double well potential

Abstract

Quantum quartic single-well anharmonic oscillator Vao(x) = x2 + g2x4 and double-well anharmonic oscillator Vdw(x) = x2(1−gx)2 are essentially one-parametric, they depend on a combination (g2ℏ). Hence, these problems are reduced to study the potentials Vao = u2 + u4 and Vdw = u2(1 − u)2, respectively. It is shown that by taking uniformly-accurate approximation for anharmonic oscillator eigenfunction Ψao(u), obtained recently, see JPA 54 (2021) 295204 [1] and arXiv 2102.04623 [2], and then forming the function Ψdw(u) = Ψao(u)±Ψao(u−1) allows to get the highly accurate approximation for both the eigenfunctions of the double-well potential and its eigenvalues.