and its derivation remain valid.

For a Coulomb potential, the square of wave function at the origin of the $n$S state, $|\phi_n(0)|^2=u'_n(0)^2$, decreases like $1/n^3$, and so does the energy shift, a property which is well known for exotic atoms.

The $n$-dependence of $|\phi_n(0)|^2$ has been discussed, e.g., in the context of charmonium physics [@Quigg:1979vr; @Grosse:1997xu]. For power-law potentials $\epsilon(\alpha)r^\alpha$ ($\epsilon$ is the sign function), $|\phi_n(0)|^2$ increases with $n$ if $\alpha<1$, and decreases if $\alpha>1$. If $\alpha=1$, then $|\phi_n(0)|^2$ is independent of $n$ (after normalisation). This can be seen from the Schwinger formula [@Quigg:1979vr; @Grosse:1997xu] $$u'(0)^2=\int_0^\infty V'(r) u^2(r)\mathrm{d} r~,$$ which is also useful for numerical calculations.

For the harmonic oscillator (rescaled to $-u''(r)+ r^2 u(r)=E\, u(r)$ for S waves), it can be shown that $$u_n'(0)^2 = 1/B(3/2,3/2+n/2)\sim \sqrt{n}~,$$ in terms of the Euler function $B(x,y)= \Gamma(x) \Gamma(y)/\Gamma(x+y)$.

Note that the question of a large $n$ limit has a different answer for the Coulomb and oscillator cases. In the former case, the $n$-S radial wave function $u_n(r)$ extends outside when $n$ increases, with an asymptotic decrease (in our normalisation) $\exp(-r/(2 n)$. The $2$S state is $u_2(r)\propto r(4-r)\exp(-r/4)$ has its first (an unique) node at $r_1(2)=4$. As $n$ increases, this first node 