Abstract
We establish an asymptotic representation of the function \(\tilde n(R) = \int\limits_0^R {\frac{{n(r) - n(0)}}{r}dr, R \in \Re } \subseteq [0, \infty ), R \to \infty ,\) where n(r) is the number of eigenvalues of the Sturm-Liouville problem on [0,∞) in (λ:¦λ¦≤r) (counting multiplicities). This result is obtained under assumption that q(x) slowly (not faster than In x) increases to infinity as x→∞ and satisfies additional requirements on some intervals \([x_ - (R), x_ + (R)],R \in \Re \).
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Palyutkin, V.G. Distribution of eigenvalues of the Sturm-Liouville problem with slowly increasing potential. Ukr Math J 48, 914–927 (1996). https://doi.org/10.1007/BF02384176
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DOI: https://doi.org/10.1007/BF02384176