From quartic anharmonic oscillator to double well potential

Abstract

It is already known that the quantum quartic single-well anharmonic oscillator $V_{ao}(x)=x^2+g^2{x}^4$ and double-well anharmonic oscillator $V_{dw}(x)=x^2(1-gx{)}^2$ are essentially one-parametric, their eigenstates depend on a combination $(g^2\hslash )$. Hence, these problems are reduced to study the potentials $V_{ao}=u^2+u^4$ and $V_{dw}=u^2(1-u{)}^2$, respectively. It is shown that by taking uniformly-accurate approximation for anharmonic oscillator eigenfunction ${\Psi }_{ao}(u)$, obtained recently, see JPA 54 (2021) 295204 [1] and Arxiv 2102.04623 [2], and then forming the function ${\Psi }_{dw}(u)={\Psi }_{ao}(u)\pm {\Psi }_{ao}(u-1)$ allows to get the highly accurate approximation for both the eigenfunctions of the double-well potential and its eigenvalues.

7 pages, extended, two figures added, to be published at Acta Polytechnica