From quartic anharmonic oscillator to double well potential

Abstract

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