A3 - 1D simple harmonic oscillator perturbed by a central potential

V_{0} \exp (- \frac{x^{2}}{ε})

\begin{aligned} E_{0}^{1} & = \int_{- \infty}^{\infty} {(\frac{1}{\sqrt{π} b})}^{\frac{1}{2}} \exp (- \frac{x^{2}}{2 b^{2}}) V_{0} \exp (\frac{- x^{2}}{ε}) {(\frac{1}{\sqrt{π} b})}^{\frac{1}{2}} \exp (- \frac{x^{2}}{2 b^{2}}) d x \\ = \frac{V_{0}}{{(1 + \frac{b^{2}}{ε})}^{\frac{1}{2}}} \end{aligned}