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We show that, in the presence of a scalar field the range of the value of external field parameters a and b, at which corresponding Hamiltonian operator is hermitian, essentially wider than in its absence. It allows us to study precisely the question on stability of QED vacuum in the presence of a strong electric field of a point charge Z|e| and external scalar Coulomb field with respect to electron-positron pair production. Also, we consider the scattering of Dirac particle by the specified fields in 3 + 1 dimensions. The phase shift and wave functions are obtained exactly. We calculate the scattering amplitude in a quasiclassical approximation as a partial wave series. By means of figures obtained for the cross section σ (θ) in general and special cases, such as a = b and a = b, we find that σ (θ) is not exactly symmetric about θ = π.

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