Exact analytical expression of the Fredholm determinant with outgoing wave boundary condition for motion in Hulthén-distorted non-local separable potential is constructed and expressed in the maximum reduced form. Using boundary conditions (regular and irregular), two approximate energy-dependent interactions corresponding to the parent non-local potential are also constructed. The phase shifts for the $\alpha–\alpha$ elastic scattering are computed by using (i) exact expression for the Fredholm determinant and (ii) energy-dependent local interactions by exploiting the phase function method. The merits of our constructed equivalent energy-dependent potentials are judged by comparing the $\alpha–\alpha$ elastic scattering phases with our exact calculation and standard data.

By exploiting higher partial wave solutions for the Hulthén potential, constructed via the factorisation method, closed form analytical expressions of the Fredholm determinants for motion in Hulthén plus modified Graz separable potential are constructed to study on-shell scattering up to partial wave $\scr l$= 2. Phase shifts for different states of $\alpha-^3$H and $\alpha-^3$He are obtained by exploiting the expression of the Fredholm determinant. The results are found in reasonable agreement with the standard data (Spiger and Tombrello 1967).

An equivalent energy-dependent local potential corresponding to Coulomb plus Graz separable potential is constructed through simple rearrangement of the Schrödinger equation. It is conjectured that local Coulomb-like potential is equally applicable for the traditional phase function method. The merit of our constructed potential is thus judged by studying nucleon–nucleon and alpha–nucleon systems through the phase function method. Good agreement in phase shift values with standard data is achieved.

New analytical expressions for the off-shell wave functions and T-matrix with the Manning–Rosen potential are constructed in terms of generalised hypergeometric functions. The off-shell T-matrices are computed for the neutron–proton and neutron–deuteron systems. The limiting behaviours of our expression for the off-shellT-matrix are verified and found correct.

The scattering phase parameters for short-range local potentials can be evaluated by using the phase function method (PFM), regarded as an efficient approach for computing scattering phase shifts for quantum mechanical problems, without solving the Schrödinger equation. We adapt PFM to deal with the Hulthén-distorted separable non-local potentials and derive a closed form expression for the phase function with rigorous inclusion of electromagnetic effect. We demonstrate the usefulness of our constructed expression by calculating elastic scattering phase parameters for proton–deuteron (p–d) system which agree quite well with the previous results.