Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2019 / Paper 1 / 10 / d / Solution

For any real coupling, the same normalized Gaussian family has energyIf $ϵ<0$, then as $\alpha \downarrow 0$ the negative ${\alpha }^{-2}$ term dominates the positive ${\alpha }^{-1}$ and $\alpha$ terms. Each trial still has finite energy, is normalized and decays at infinity, but its energy tends to $-\infty$. ThereforeThis variational sequence makes the instability of the negative quartic potential explicit, without mistaking any local well near the origin for a global ground-state bound.