We have considered the stationary state of a one-dimensional Gaussian polymer chain of length Nl (N is number of segments, l is the Kuhn length) subjected to a one-parameter class of exactly solvable, symmetric, repulsive potentials with multiple (two or three) minima. We have calculated analytically and numerically the exact Green's function G(R\R-';L) and the mean-square end-to-end distance [(R-R-')(2)], respectively, of the polymer chain. The instanton approximation is translated in polymer physics language and used to analyze the conformation of the polymer chain in its ground state, by evaluating the average number of folds that connect the potential minima. All quantities are functions of the barrier height and of the separation between wells. Our results show that for a given length N, the polymer expands with increasing barrier width until a maximum value of [(R-R-')(2)] is reached. Afterwards, the polymer apparently collapses in one of the wells. This behavior defines a critical length N* and may offer the possibility of applications in separation and pattern recognition processes.