Calculation of the critical bound–continuum limit of the one-electron atoms interacting with the generalised exponential cosine screened potential using scattering formalism

Abstract The critical values of the screening parameters ( β and θ ) of the generalised exponential cosine screened potential (GECSCP), characterising the critical bound–continuum limit of the one-electron atoms interacting with this potential, have been calculated accurately by exploring the scattering state solution of the corresponding Schrödinger equation at the zero-energy. It is shown that the scattering length (SL) corresponding to an attractive potential becomes singular and the zero-energy resonance occurs when the potential loses the ability of supporting any bound state. More generally, when the potential loses the ability of holding a particular s bound state, the SL becomes singular. In order to obtain the SL, correct up to the twelfth decimal place, for the attractive GECSCP as a function of β and θ , the corresponding Schrödinger equation is solved numerically within the framework of the Numerov method containing an error of order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>0.1</mml:mn> <mml:mo>×</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>16</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . The quantitative and qualitative natures of the SL are then analysed in details to obtain the accurate critical values of the screening parameters that correspond to the singularities of the SL. Special emphasis is given to study the critical behaviour of the cosine Coulomb potential. Our present results for the critical bound–continuum limit are in nice agreement with the hitherto established results.