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The radial Schrödinger equation with an attractive Gaussian potential and a general angular momentum is transformed by means of the modified Laplace transformation into a linear homogeneous differential equation of the first order with one "retarded" argument. Owing to the fusion of the arguments at the point $z=0$ its integration is possible by an iteration procedure. The discrete spectrum differs from the continuous one by the boundary condition at $z=\infty$ which determines the explicit equation for the energy eigenvalues. The properties of the resolvent are investigated in detail on the real half-axis and various approximations are dicussed.
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