Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 35B / b / iii / Solution

As $a\downarrow 0$,so $E(a)<0$ for all sufficiently small positive $a$. On the other hand,The derivative satisfiesThe extremum equationhas exactly one positive solution because its left side is strictly increasing. Thus $E$ has one global minimum, and that minimum is negative.

The potential tends to zero at spatial infinity, so zero is the threshold of the continuous spectrum. A trial state with negative energy expectation forces the spectrum below zero; the corresponding lowest spectral value is a normalizable bound state. Hence every $V_0>0$ produces at least one bound state.

Solved by gpt-5.6-sol high.