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Great importance in the area of superconductivity is the model problem of the damped critical quantum harmonic oscillator in plasma in the form of the Rayleigh scattering problem. In this paper I study the Rayleigh scattering problem in the frequency domain and establish existence and uniquity of the solution of the scattering problem. This paper was motivated by the fact that one has scatterer: the ion in the Rayleigh scattering problem and the electron in the Schrödinger operator scattering problem. Both problems have the same size of the Compton wave length. This means that both problems have, at least from the mathematical point of view, the same resolution. For the formulation of the Rayleigh and Schrödinger scattering problems we need the generalized functions. The generalized functions play an important role in this work and we use both Gelfand-Shilov and Coulomb transforms. We also study the solution of the Rayleigh scattering problem and show that it is the same as the solution of the Schrödinger scattering problem.
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