Geodesic
The substantiation of the method of two-sided approximations for eigenvalues of the second-order elliptic operator
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 4 (2001) no. 1, pp. 61-84

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In the paper, the method of two-sided approximation for the spectral problems with the second-order elliptic operator based on the well-known fictitious method is substantiated. The expansion of eigenvalues of auxiliary problem in power series by small parameter of continuation irrespective of its sign is proved. Unimprovable error bounds of two-sided approximations are obtained. The conjugate-factorized structure of the problem operator plays the decisive role in obtaining the result. The research is realized on discrete level. 
@article{SJVM_2001_4_1_a6,
     author = {S. B. Sorokin},
     title = {The substantiation of the method of two-sided approximations for eigenvalues of the second-order elliptic operator},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {61--84},
     publisher = {mathdoc},
     volume = {4},
     number = {1},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2001_4_1_a6/}
}
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S. B. Sorokin. The substantiation of the method of two-sided approximations for eigenvalues of the second-order elliptic operator. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 4 (2001) no. 1, pp. 61-84. http://geodesic.mathdoc.fr/item/SJVM_2001_4_1_a6/