Geodesic
Effective criteria for the~strong sign-regularity and the~oscillation property of the~Green's functions of two-point boundary-value problems
Sbornik. Mathematics, Tome 188 (1997) no. 11, pp. 1687-1728

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Necessary and sufficient conditions for strong sign-regularity and the oscillation property (in the sense of Gantmakher and Krein) of the Green's function of a two-point boundary eigenvalue problem are obtained. These conditions guarantee that even in a non-self-adjoint case the eigenvalues are real and have several other spectral properties similar to those of the classical Sturm–Liouville problem. The conditions are formulated in terms of the properties of a uniquely defined fundamental system of solutions of the differential equation. This makes it possible to verify them effectively using a computer and to establish, as the final result, the oscillation property of the Green's function and the corresponding spectral properties of the boundary-value problem in a large number of cases in which these properties could not be detected on the basis of previously known sufficient conditions. 
@article{SM_1997_188_11_a4,
     author = {G. D. Stepanov},
     title = {Effective criteria for the~strong sign-regularity and the~oscillation property of {the~Green's} functions of two-point boundary-value problems},
     journal = {Sbornik. Mathematics},
     pages = {1687--1728},
     publisher = {mathdoc},
     volume = {188},
     number = {11},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1997_188_11_a4/}
}
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G. D. Stepanov. Effective criteria for the~strong sign-regularity and the~oscillation property of the~Green's functions of two-point boundary-value problems. Sbornik. Mathematics, Tome 188 (1997) no. 11, pp. 1687-1728. http://geodesic.mathdoc.fr/item/SM_1997_188_11_a4/