The spectrum of the boundary problems related to the double confluent case of Heun's differential equation is studied numerically and by means of asymptotic methods. The calculation is based on an application of the central two-point connection problem for this equation using Jaffé expansions and Birkhoff sets of irregular difference equations of Poincaré-Perron type. The numerical evaluation based on this approach is compared with results of asymptotic calculations showing several quite interesting features of the eigenvalue curves and of the solution of the equation itself.