Numerical study of energy-level crossing

We use a rapidly convergent numerical method recently proposed by Graffi and Grecchi to study the analytic properties of the eigenvalues $E\left(\epsilon \right)$ of the differential equation $[-(\frac{d^2}{d{x}^2})+\epsilon \mathrm{}|x|+\frac{1}{4}{x}^2-E(\epsilon \left)\right]\psi \mathrm{}\left(x\right)=0\mathrm{}$, ${\lim }_{\mathrm{}\left|x\right|\mathrm{}\to \infty }\psi \mathrm{}\left(x\right)=0\mathrm{}$. We analytically continue $E\left(\epsilon \right)$ into the complex $\epsilon$ plane on a computer, demonstrate graphically the phenomenon of level crossing, and show that the crossing points are square-root branch points.