Geodesic
A resolvent approach in the Fourier method for the wave equation: The non-selfadjoint case
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 7, pp. 1156-1167

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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. 
@article{ZVMMF_2015_55_7_a5,
     author = {V. V. Kornev and A. P. Khromov},
     title = {A resolvent approach in the {Fourier} method for the wave equation: {The~non-selfadjoint} case},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1156--1167},
     publisher = {mathdoc},
     volume = {55},
     number = {7},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_7_a5/}
}
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V. V. Kornev; A. P. Khromov. A resolvent approach in the Fourier method for the wave equation: The non-selfadjoint case. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 7, pp. 1156-1167. http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_7_a5/