Asymptotology - A cautionary tale

The art of asymptotology is a powerful tool in applied mathematics and theoretical physics, but can lead to erroneous conclusions if misapplied. A seemingly paradoxical case is presented in which a local analysis of an exactly solvable problem appears to find solutions to an eigenvalue problem over a continuous range of the eigenvalue, whereas the spectrum is known to be discrete. The resolution of the paradox involves the Stokes phenomenon. The example illustrates two of Kruskal's Principles of Asymptotology.