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The paper is devoted to the proof of the uniqueness theorem for solution ofthe equation for the non-local ionization source in a glow discharge and ahollow cathode in general 3D geometry. The theorem is applied to wide class ofelectric field configurations, and to the walls of discharge volume, which havea property of incomplete absorption of the electrons. Cathode is regarded asinterior singular source, which is placed arbitrarily close to the wall. Theexistence of solution is considered also. During the proof of the theorem manyof useful structure formulae are obtained. Elements of the proof structure,which have arisen, are found to have physical sense. It makes clear physicalconstruction of non-local electron avalanche, which builds a source ofionization in glow discharge at low pressures. Last has decisive significanceto understand the hollow cathode discharge configuration and the hollow cathodeeffect.

Uniqueness theorem for the non-local ionization source in glow discharge and hollow cathode