= Solution
Let $z=E-V_0+i0$. The <outgoing analytic branch of Klein-Gordon step momentum> is
$$
\boxed{q(z)=\frac{\sqrt{z-M}\sqrt{z+M}}{\hbar c}},
$$
where each <square root> is continued from the <upper half-plane>. It behaves as $z/(\hbar c)$ at large positive or negative real $z$. The <branch points> are $E=V_0\pm M$; replacing this product by a principal <square root> of $z^2-M^2$ can choose the wrong sign below the lower threshold.
\b[I.] For $E-V_0>M$, $q>0$ and both transmitted <kinetic energy> and <group velocity> are positive. Thus $0\leq\mathcal R<1$ and $\mathcal T=4kq/(k+q)^2>0$; the transmission decreases to zero on approaching the upper threshold from above.
\b[II.] For $|E-V_0|<M$, write $q=i\kappa$ with $\kappa=\sqrt{M^2-(E-V_0)^2}/(\hbar c)>0$. The right wave decays, $r=(k-i\kappa)/(k+i\kappa)$ has modulus one, and $\mathcal T=0$. There is an evanescent tail but no <flux> transmitted to infinity. Its reflection <quantum phase> changes continuously between the two thresholds.
\b[III.] For $E-V_0<-M$, the <analytic continuation> gives $q=-Q$, where $Q=\sqrt{(E-V_0)^2-M^2}/(\hbar c)>0$. The <group velocity> $c^2\hbar q/(E-V_0)$ is again rightward, while the charge density and transmitted charge current are negative. Therefore
$$
\boxed{\mathcal R=\frac{(k+Q)^2}{(k-Q)^2}>1,\qquad
\mathcal T=-\frac{4kQ}{(k-Q)^2}<0,\qquad
\mathcal R+\mathcal T=1}.
$$
Choosing $q=+Q$ would describe a wavepacket arriving from the right, inconsistent with the chosen incoming channel. As the lower threshold is approached, $q\to0^-$ and $\mathcal R\to1$ from above. This <Klein-Gordon step amplification> has a real denominator zero at $E=V_0/2$ when $V_0>2M$: there $Q=k$, and no finite stationary solution with unit incoming amplitude satisfies both matching equations. The resulting formal pole must not be described as an infinite single-particle <probability>. Finite switching, wavepackets and the treatment of the external source require a multiparticle analysis.<Klein Paradox for Spin-0 Particles by Douglas M. Gingrich>[Equations 4.142–4.155: outgoing negative-momentum branch, boundary matching and charge currents]{ref}
\Image[/past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2013/paper-1-klein-step-branches.png]
{title=Original outgoing Klein-Gordon momentum branches and charge-current reflection for a supercritical sharp step}
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