Geodesic
Asymptotics of the eigenvalues of the Dirichlet and Neumann problems for the abstract Sturm-Liouville operator
Matematičeskie zametki, Tome 21 (1977) no. 2, pp. 209-212

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Let $A>0$ be an unbounded self-adjoint operator in a Hilbert space $H$. In the Hilbert space $H_1=L_2(0,\pi;H)$ we study the spectrum of the differential equations \begin{gather*} -y''(x)+Ay=\lambda y,\quad y(0)=y(\pi)=0, \\ -y''(x)+Ay=\lambda y,\quad y'(0)=y'(\pi)=0. \end{gather*} We find the principal terms of the asymptotics of the functions $N(\lambda)$ for these problems and we ascertain the conditions under which they are asymptotically not equivalent. 
@article{MZM_1977_21_2_a8,
     author = {M. M. Gekhtman},
     title = {Asymptotics of the eigenvalues of the {Dirichlet} and {Neumann} problems for the abstract {Sturm-Liouville} operator},
     journal = {Matemati\v{c}eskie zametki},
     pages = {209--212},
     publisher = {mathdoc},
     volume = {21},
     number = {2},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1977_21_2_a8/}
}
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M. M. Gekhtman. Asymptotics of the eigenvalues of the Dirichlet and Neumann problems for the abstract Sturm-Liouville operator. Matematičeskie zametki, Tome 21 (1977) no. 2, pp. 209-212. http://geodesic.mathdoc.fr/item/MZM_1977_21_2_a8/