For the example of the nonrelativistic Schrödinger operator, methods are formulated for calculating the determinant of an elliptic operator on the basis of scattering theory. It is shown that such a determinant is identical to the Jost determinant at zero energy. In the centrally symmetric case, it reduces to ordinary Jost functions and ultimately to the values of the zero-energy wave functions at the origin. The relationship between the determinant of the Schrödinger operator and the characteristics of the scattering resonances and the number of bound states in a field of opposite sign is noted. This makes it possible to find the first terms in the gradient expansion of the determinant as a functional of the potential. The problem of the correlation free energy of a classical plasma serves as a physical illustration.