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In the paper the closed Friedmann--Robertson--Walker model with quantizationin the presence of the positive cosmological constant and radiation is studied.For analysis of tunneling probability for birth of an asymptotically deSitter,inflationary Universe as a function of the radiation energy a new definition ofa "free" wave propagating inside strong fields is proposed. On such a basis,tunneling boundary condition is corrected, penetrability and reflectionconcerning to the barrier are calculated in fully quantum stationary approach.For the first time non-zero interference between the incident and reflectedwaves has been taken into account which turns out to play important role insidecosmological potentials and could be explained by non-locality of barriers inquantum mechanics. Inside whole region of energy of radiation the tunnelingprobability for the birth of the inflationary Universe is found to be close toits value obtained in semiclassical approach. The reflection from the barrieris determined for the first time (which is essentially differs on 1 at theenergy of radiation close to the barrier height). The proposed method could beeasily generalized on the cosmological models with the barriers of arbitraryshape, that has been demonstrated for the FRW-model with included Chaplygingas. Result is stable for variations of the studied barriers, accuracy arefound to be 11--18 digits for all coefficients and energies below the barrierheight.

A fully quantum method of determination of penetrability and reflection coefficients in quantum FRW model with radiation