ANALYTIC CONTINUATION OF EIGENVALUE PROBLEMS

In this paper we consider the dependence of Schrodinger equation eigenvalue problems on the coupling-constant parameters in the potential. We show that unless great care is taken, analytic continuation in these parameters can lead to surprising and paradoxical conclusions. Careful analysis of the analytical properties shows that such eigenvalue problems can have elaborate internal structure; these problems can incorporate several different eigenvalue problems joined together. For example, the anharmonic oscillator, whose potential is V(x) = a2x6 - 3ax2, is actually four different eigenvalue problems combined into one.