A resolvent approach in the Fourier method for the wave equation: The non-selfadjoint case

Under minimum smoothness requirements for the initial data, the Fourier method in the mixed problem for the wave equation with a complex potential is justified by using the Cauchy–Poincare technique for the contour integration of the resolvent of the eigenvalue problem. Generic boundary conditions are used; one of them contains first-order derivatives, while the other does not. In this case, even for the benchmark situation, the operator in the eigenvalue problem can have any number of generalized eigenfunctions. A substantial use is made of the technique for accelerating Fourier series due to A. N. Krylov.