quantum mechanics - A problem with the Gamow state

Consider a form of potential $U(r)$ as follows $$ U(r)=\begin{cases}0 & 0b\end{cases} $$

In this problem $r$ is the distance from the origin, $r \geq 0$.

It is known that the Schrodinger equation $$ \left[\frac{\hat P^2}{2m} + U(r) - E\right] \psi (r) = 0 $$ admits two types of solutions: real energy solutions, with $E \geq 0$ (also named "scattering states", and complex energy solutions, with $E = E_0 - i\Gamma$ ("resonant states").

From physical reasons, both these solutions have to satisfy $\psi (r=0) = 0$.

I ask opinion about two statements.

1. 
   

   The scattering states ($E$ real and positive) form a complete basis of functions according to which we can develop in a quantum superposition, any function that vanishes at $r = 0$.

   
   



2. 
   

   Therefore, the Gamow state (complex $E$) can be expanded in this basis, i.e. the Gamow state is equal to its expansion in this basis.

   
   

Note: keep in mind that the scattering states are normalized to the Dirac $\delta$,

$$ \int dr \ \psi^*(r;k) \psi (r; k') = \delta (k - k'), $$

and the Gamow state diverges for $r \to \infty $.

I know that the 1st statement is correct. My question on these statements is because of things that I see quite often in the literature: the energy spectrum of a well isolated resonance, narrow and far from the threshold, is said to be distributed Breit-Wigner. But, for finding the spectrum of real energies one needs the above expansion of the resonance state, i.e. as a quantum superposition of real-energy states. But those who claim the Breit-Wigner distribution, don't do the expansion in a quantum superposition. This is what leads me to my questions.