From quartic anharmonic oscillator to double well potential

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 $(g^2 \hbar)$. 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.


I. INTRODUCTION
For the one-dimensional quantum quartic single-well anharmonic oscillator V ao (x) = x 2 + g 2 x 4 and double-well anharmonic oscillator with potential V dw (x) = x 2 (1 − gx) 2 the (trans)series in the coupling constant g (which is the Perturbation Theory in powers of g (the Taylor expansion) in the former case of V ao (x) supplemented by exponentially-small terms in g in the latter case of V dw (x)) and the semiclassical expansion in (the Taylor expansion for V ao (x) supplemented by the exponentially small terms in for V dw (x)) for energies coincide [3].This property plays crucially important role in our consideration.
Both the quartic anharmonic oscillator with a single harmonic well at x = 0 and the double-well potential with two symmetric harmonic wells at x = 0 and x = 1/g, respectively, are particular cases of the quartic polynomial potential where g is the coupling constant and a is a parameter.Interestingly, the potential (3) is symmetric for three particular values of the parameter a: a = 0 and a = ±2.All three potentials (1), ( 2), (3) belong to the family of potentials of the form for which there exists a remarkable property: the Schrödinger equation becomes oneparametric, both the Planck constant and the coupling constant g appear in the combination ( g 2 ), see [2].It can be immediately seen if instead of the coordinate x the so-called classical coordinate u = (g x) is introduced.This property implies that the action S in the path integral formalism becomes g-independent and the factor 1 in the exponent becomes 1 g 2 [4].Formally, the potentials ( 1)-( 2), which enter to the action, appear at g = 1, hence, in the form respectively.Both potentials are symmetric with respect to u = 0 and u = 1/2, respectively.
Namely, this form of the potentials will be used in this short Note.This Note is the extended version of a part of presentation in AAMP-18 given by the first author [5].

II. SINGLE-WELL POTENTIAL
In [1] for the potential (4) matching the small distances u → 0 expansion and the large distances u → ∞ expansion (in the form of semiclassical expansion) for the phase φ in the representation of the wave function, where P is a polynomial, it was constructed the following function for the (2n + p)-excited state with quantum numbers (n, p), n = 0, 1, 2, . . ., p = 0, 1 : where P n,p is some polynomial of degree n in u 2 with positive roots.Here A = A n,p , B = B n,p are two parameters of interpolation.These parameters (−A), B are slow-growing with quantum number n at fixed p taking, in particular, the values for the ground state and the first excited state, respectively.This remarkably simple function (6), see Fig. 1 (top), provides 10-11 exact figures in energies for the first 100 eigenstates.

IV. DOUBLE-WELL POTENTIAL: RESULTS
In this section we present concrete results for energies of the ground state (0, 0) and of the first excited state (0, 1) obtained with the function (9) at p = 0, 1, respectively.The results are compared with the Lagrange-Mesh Method (LMM) [7].
A. Ground State (0,0) The ground state energy for (5) obtained variationally using the function (9) at p = 0 and compared with LMM results [7], where all printed digits (in the second line) are correct, E (0,0)