Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 4 / 33C / b

For the odd channel, solve on $x>0$ with $\psi (0)=0$. Inside $0<x<a$, write $\psi =C\sin (kx)$; outside write $\psi =e^{-ikx}-S_{--}{e}^{ikx}$, respecting the outgoing odd-channel minus sign. Continuity at $a$ and the delta-potential derivative jump areEliminating $C$ yieldsThe equations remain valid when $\sin (ka)=0$ by taking the limiting value, rather than dividing blindly. For real $k,U_0$, numerator and denominator inside the fraction are complex conjugates and the prefactor has unit modulus. Thus $|S_{--}{|}^2=1$. This is conservation of flux in the odd parity channel: the real potential causes only a phase shift there, not absorption. It does not assert unit left-to-right transmission in the usual incident-from-one-side basis.

A negative-energy odd bound state has $k=i\kappa$, $\kappa >0$, with $\psi =C\sinh (\kappa x)$ inside and a decaying exponential outside. Its jump condition, equivalently the nontrivial upper-half-plane pole of the preceding S-matrix, givesSet $w=2\kappa a$ and $g=-a{U}_0>0$. The condition is $w=g(1-e^{-w})$. The right side is strictly increasing and strictly concave, has slope $g$ at zero and is bounded by $g$. Hence there is exactly one positive intersection iff $g>1$. ThereforeThe formal zero root is not a positive-decay bound state. At $a{U}_0=-1$ there is a zero-energy threshold solution, not a normalizable bound state.

Let $g=a{U}_0\gg 1$ and $z=ka$. A denominator zero away from the removable zero at $z=0$ obeysNear $z=\pi$, write $z=\pi +\delta$ and choose the logarithm branch with $\delta \to 0$. Expanding givesConsequentlyThe lower-half-plane pole is a scattering resonance: an odd standing wave is nearly confined between the two high repulsive barriers but leaks out slowly. If $E=E_R-i\Gamma /2={\hslash }^2{k}^2/(2m)$, then $\Gamma \simeq 2{\hslash }^2\pi \gamma /(m{a}^2)$ and its probability lifetime is $\hslash /\Gamma$. It is not a genuine bound state of the repulsive potential.