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My understanding is that bound states are solutions to the Schrodinger equation where $\lim_{x\to\pm\infty} V(x)=0$ and $\min V(x)\leq E \leq 0$, in which case there are discrete allowed energies. In the scattering case, $E\geq 0$ and there are no limitations on allowed energies.

So let's say you have an arbitrary wavefunction in such a potential. Logically, it would be $\Psi(x)=\sum_{bounded} C_n\psi_{n}+\int_{k_0}^{\infty} dk e^{ikx}\phi(k)$ where $k_0$ is the first $k$ such that $E\geq0$ and $\phi(k)$ is the fourier transform of $\Psi(x)$.

Is my reasoning right? How would you go about finding $C_n$ and $\phi(k)$ for a known $\Psi(x)$ in this case? There doesn't seem to be an obvious way to use the orthogonality of solutions to find $C_n$ and $\phi(k)$?

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you are right there is no way to use orthognality and this issue is mainly due to the scattering states. The scattering states are not orthogonal to the bound states and even to each other.

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