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We give explicit local intertwining maps to multiplier representations and show that the study of the solution space for potentials of the form $V(t,x)=g_2(t)x^2+g_1(t)x+g_0(t)$ reduces to the study of the potential free case. We also show that the study of the time-dependent potentials of the form $V(t,x)=\\lambda x^{-2}+g_2(t)x^2+g_0(t)$ reduces to the study of the potential $V(t,x)=\\lambda x^{-2}$. 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