$r_1(n)$ necessarily decreases, as a consequence of the interlacing theorem, however, $\lim_{n\to\infty}\simeq 3.67$, the first node of the Bessel function which satisfies $y''+y/r=0$, $y(0)=0$. Hence if a potential is short-ranged for 1S, it is also short-ranged for all $nS$ states, and also for states with orbital momentum $\ell>0$. On the other hand, for the harmonic oscillator, all $n$S states have about the same size, with the same asymptotic fall-off $\exp(-r^2/2)$. As $n$ increases, the radial equation is approximately $u''+4n u=0$, with first node $r_1(n)\sim \pi/\sqrt{4n}$. Hence an additional potential whose range is short but finite will feel the node structure of states with very high $n$, and the approximation leading to the generalised Deser–Trueman formula (\[eq:Trueman\]) ceases to be valid. These considerations hold for an harmonic oscillator with fixed strength.

This work was done in the framework of the CEFIPRA Indo–French exchange programs 1501-2 and 3404-D: Rigorous results in quantum information theory, potential scattering and supersymmetric quantum mechanics. It is a pleasure to thank A. Martin and T.E.O. Ericson for very interesting discussions.

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Ya.B. Zel’dovich, “Energy levels in a distorted Coulomb field”, Sov. J. Solid State, [**1**]{} (1960) 1497.

E. B. Kolomeisky, and M. Timmins, *Physical 