Eigenvalues of the one-dimensional Smoluchowski equation

Abstract

The eigenvalues and eigenfunctions of the Smoluchowski equation are investigated for the case of potentials withN deep wells. The small parameter δ=kT/V, which measures the ratio of the thermal energy to a typical well depth, is used in connection with the method of matched asymptotic expansion to obtained asymptotic approximations to all the eigenvalues and eigenfunctions. It is found that the eigensolutions fall into two classes, namely (i) the top-of-the-well and (ii) the bottom-of-the-well eigensolutions. The eigenvalues for both classes of solutions are integer multiples of the squqres of the frequencies at the top or bottom of the various wells. The eigenfunctions are, in general, localized to the top or bottom of the corresponding well. The very small eigenvalues require special consideration because the asymptotic analysis is incapable of distinguishing them from the zero eigenvalue with multiplicityN. Another approximation reveals that, in addition to the true zero eigenvalue, there areN-1 eigenvalues of order exp(−δ). The case of other possible multiple eigenvalues is also examined.