Qualitative behavior of a potential based on physically acceptable wavefunction

Question

Given the behavior of a wavefunction, how should one gain an intuition about the qualitative behavior of the potential in the S.E for which this wavefunction is a physically acceptable solution?
Looking for hints!

ytlu · Answer

From the time-independ shr$ddot{o}$dinger equation:
$$
   V(x) - E = frac{1}{Psi(x)}frac{hbar^2}{2m}frac{d^2Psi(x)}{dx^2}.
$$

Qualitative behavior of a potential based on physically acceptable wavefunction

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