  The Transfer-Integral (TI) operator is a powerful method to investigate the
statistical physics of 1-dimensional models, like those used to describe DNA
denaturation. At the cost of a certain number of approximations, the TI
equation can be reduced to a Pseudo-Schr\"odinger Equation (PSE), according to
which the DNA sequence is equivalent to a point particle moving in a potential
well. In this paper, I check the validity of the standard PSE approximation for
two different 1-dimensional DNA models, and show that it fails to provide
correct results for both of them. I then propose a generalized PSE, which works
well for one of the two models. Finally, I discuss the particle description of
DNA denaturation that is derived from this generalized PSE.
