concave function of $\lambda$, and monotonic if $v(r)<0$ [@Thirring:1979b3]).

Beyond the critical region $\lambda\sim \lambda_1$, the scattering length $a$ becomes small again, but *positive*. Remarkably, the Deser–Trueman formula (\[eq:Trueman\]) is again valid, and accounts for the nearly horizontal plateau experienced by the second state near $E_{0,1}=-1/4$. A spectroscopic study near $\lambda\gs \lambda_1$ would reveal a sequence of seemingly 1S, 2S, 3S, etc., states slightly shifted *upwards* though the Coulomb potential is modified by an *attractive* term. This is intimately connected with very low energy scattering: a negative phase-shift $\delta$ can be observed with an attractive potential which has a weakly-bound state, and mimics the effect of a repulsive potential. (The difference will manifest itself if energy increases: the phase-shift produced by a repulsive potential will evolve as $\delta(T)\to 0$ as the scattering energy $T$ increases, while for the attractive potential with a bound state, according to the Levinson theorem, $\delta(T)\to -\pi$.)

The occurrence of an atomic level near $E_{0,1}=-1/4$ for $\lambda\gs \lambda_1$ can also be understood from the nodal structure. A deeply-bound nuclear state has a short spatial extension, of the order $b$. To ensure orthogonality with this nuclear state, the first atomic state should develop an oscillation at short 