Geodesic
Completeness of systems of eigenfunctions for the~Sturm--Liouville operator with potential depending on the~spectral parameter and for one non-linear problem
Sbornik. Mathematics, Tome 188 (1997) no. 7, pp. 1071-1084

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The eigenvalue problem for the Sturm–Liouville operator on the closed interval $[0,1]$ with potential depending on the spectral parameter and with zero Dirichlet boundary conditions is considered first. It is proved under certain assumptions about the potential that if a system of eigenfunctions of this problem contains a unique function with $n$ zeros in the interval $(0,1)$ for each non-negative integer $n$, then it is complete in the space $L_2(0,1)$ if and only if the functions in this system are linearly independent in $L_2(0,1)$. Next, this result is used in the study of the spectral problem for a certain non-linear operator of Sturm–Liouville type. The completeness in $L_2(0,1)$ of the corresponding eigenfunctions is proved. 
@article{SM_1997_188_7_a5,
     author = {P. E. Zhidkov},
     title = {Completeness of systems of eigenfunctions for {the~Sturm--Liouville} operator with potential depending on the~spectral parameter and for one non-linear problem},
     journal = {Sbornik. Mathematics},
     pages = {1071--1084},
     publisher = {mathdoc},
     volume = {188},
     number = {7},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1997_188_7_a5/}
}
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P. E. Zhidkov. Completeness of systems of eigenfunctions for the~Sturm--Liouville operator with potential depending on the~spectral parameter and for one non-linear problem. Sbornik. Mathematics, Tome 188 (1997) no. 7, pp. 1071-1084. http://geodesic.mathdoc.fr/item/SM_1997_188_7_a5/