Localized States near a Short-Range Defect in the Presence of a Hyperbolic Potential Field

The exact analytical solution of the one-dimensional stationary Schrödinger equation with a spatially distributed hyperbolic potential profile ~1/x in the presence of a Dirac delta function (point) potential is found and analyzed. Localization features are described analytically in dependence on the parameters of a hyperbolic potential profile and point potential. It is found that the localization energy of the ground state monotonically decreases with an increase in defect intensity. The localized states exist near the both attractive and repulsive defects. Excited states are characterized by the formation of oscillations in the region of action of the hyperbolic field. The height of the maximum of the wave function decreases with decreasing intensity of the repulsive defect and with increasing absolute value of the defect intensity in the case of attractive defect. Localization length in the hyperbolic well is greater than in a halfspace with a constant potential.