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Semitransparent One Dimensional Potential, A Green's Function Approach

We study the unstable harmonic oscillator and the unstable linear potential in the presence of the point potential, which is the superposition of the Dirac $δ(x)$ and the derivative $δ'(x)$. Using the \textit{physical} boundary conditions for the Green's function we derive for both systems the resonance poles and the resonance wave functions. The matching conditions for the resonance wave functions coincide with those obtained by the self-adjoint extensions of the point potentials and also by the modelling of the $δ'(x)$. We find that, with our definitions, the pure $bδ'(x)$ barrier is semi-transparent \textit{independent} of the value of $b$.