= Solution
For a finite step in a one-dimensional potential and constant <mass>, the stationary <Schrödinger equation> requires continuity of the <wavefunction> and its first derivative. A discontinuous derivative would introduce a delta-function term in its second derivative, unsupported by a merely finite step. A discontinuous <wavefunction> is still more singular. A delta-function potential has a different derivative jump condition; an infinite wall instead imposes zero <wavefunction> and does not require matching to an exterior zero derivative.
For a bound state of the half-well, $-V_0<E<0$. The wall at zero excludes a cosine term, and square integrability excludes exponential growth outside. Write
$$
\psi(x)=\begin{cases}A\sin kx,&0<x<a,\\B e^{-q(x-a)},&x>a,\end{cases}
\qquad
k^2=\frac{2m(E+V_0)}{\hbar^2},\quad q^2=\frac{-2mE}{\hbar^2}.
$$
Continuity at $a$ gives $A\sin ka=B$ and $Ak\cos ka=-qB$. Hence
$$
\boxed{k\cot ka=-q.}
$$
Put $z=ka$, $w=qa$, and $\nu=a\sqrt{2mV_0}/\hbar$. Matching becomes $z\cot z=-w$ with $z^2+w^2=\nu^2$, $w>0$. The negative branches of $z\cot z$ occur on $((n-1/2)\pi,n\pi)$. There is one bound-state intersection in the $n$th interval exactly when
$$
\nu>(n-\tfrac12)\pi.
$$
At equality the new level has $q=0$, so its outside tail is constant and not normalizable: it is a threshold state, not a bound state. Thus no bound state exists for $\nu\le\pi/2$, including the strict inequality asked in the question.
Exactly two negative-energy bound states require the second threshold to have been crossed and the third not crossed:
$$
\boxed{\frac{3\pi}{2}<\nu\le\frac{5\pi}{2},\qquad
9<\frac{8mV_0a^2}{\pi^2\hbar^2}\le25.}
$$
The lower endpoint is excluded and the upper endpoint included precisely because a zero-energy threshold state does not count as bound. This <half-line finite-well bound-state count> differs from an unrestricted symmetric one-dimensional well, which has no hard wall at its centre.
\Image[/past-exam-of-the-physics-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1-half-well-bound-states.png]
{title=Two bound-state intersections for a hard-wall finite well of dimensionless depth two pi}
{description=Original matching-curve illustration with a chosen depth. The graph uses only negative cotangent branches and marks the two normalizable intersections.}
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