Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 36B / b / Solution

Use the normalized scaled trial stateIfthen scaling derivatives and changing variables giveIf a finite ground state existed, $\lambda =1$ would minimize this expression, and henceIt would follow thatfor $n>2$. But a bound state of an attractive potential tending to zero at infinity must have negative energy. This is a contradiction.

Indeed, the scaling gives the stronger collapse statementwhen $n>2$. Thus the ideal singular Hamiltonian is not bounded below and has no finite ground bound state.