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The work is concerned with the Transfer Matrix Method for solving the steady-state Schrödinger equation applied for a unified evaluation of the emission current density from non-refractory cathodes. The method is applicable to arbitrary shapes of the potential barrier and its transmission probability is obtained without any analytical approximations. The Fermi-Dirac distribution for the free electrons in the metal is considered as a supply function. The results, obtained for a work function of the cathode material of 4.5 eV over a wide range of values of the surface temperature and the electric field strength, clearly show a growing deviation from those obtained by the classical Jeffreys-Wentzel-Kramers-Brillouin approximation with the increase of the electric field strength. Preliminary results are obtained to demonstrate the applicability of the Transfer Matrix method to the evaluation of the ion-assisted electron emission. A significant local enhancement of the emission current density is obtained as a result of the presence of an ion at a fixed position near the metal surface. The effect becomes very strongly pronounced at an appropriate value of the electric field strength, for which a resonant ion contribution appears.