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GENERALIZED PICARD PRINCIPLE FOR SCHRDINGER EQUATIONS WITH NEGATIVE PLANAR POTENTIALS

Let β be a compact set with null capacity in the disc Ωα = {z ||z| a} on the complex plane. Consider a stationary Schrdinger equation (-△+u) u =0, where the potential u≤0 is a Radon measure of Kato Class on Ωβα=Ωa \β. Denote by u H the class of all functions u on Ω- βα with the following properties: u is nonnegative and continuous on Ω- βα ; u vanishes on {z||z| =a};u is a distributional solution of the equation on Ω- βα . For an ideal boundary point ξ∈βofΩβαin Kerekjato-Stoilow's sense, a linear operator πζ of μH→μH is defined, and a subclass Hζ={u∈μH|πζ(μ)≡0}ofμH is introduced. It is proved that the μ-generalized Picard principle is valid for ζ on Ωβα,i.e., one of the following statements is true: (i) μH is of dimension 1, and (ii) μH/Hζ is of dimension 1.