Modeling of Double-Well Potentials for the Schrödinger Equation

A new method is proposed for determining level splitting Δ in a double-well 1D potential. Two “partner” functions (one symmetric Ψ₊ and the other antisymmetric Ψ_–) are determined. From these functions, potentials V₊(x) and V_–(x) and energies E+0">$E_{+}^0$ and E−1">$E_{-}^1$ corresponding to them are determined from the Schrödinger equation. A unique property of Ψ₊ and Ψ_– is identity E+0">$E_{+}^0$ = E−1">$E_{-}^1$, which makes it possible to determine Δ from the perturbation theory in parameter V₊(x) – V_–(x). For a double-well oscillator potential, the expression for the level splitting, which connects the instanton and single-well limits, is obtained. These results can be employed in the field theory, for which the possibility of obtaining instanton solutions from perturbation theory has been discussed more than once. A number of potentials are considered, for which the value of Δ can be determined without using the semiclassical approximation. Singular potentials of the funnel type are analyzed. The value of Δ determined in this study is compared with the results of numerical solution of the Schrödinger equation for the instanton potential.