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The distribution of nonprincipal eigenvalues of singular second-order linear ordinary differential equations. (English) Zbl 1144.34392

Author’s summary: We obtain the asymptotic distribution of the nonprincipal eigenvalues associated with the singular problem
\[ x^{\prime\prime }+\lambda q(t)x=0 \]
on an infinite interval \([a,+\infty )\). Similarly to the regular eigenvalue problem on compact intervals, we can prove a Weyl-type expansion of the eigenvalue counting function, and we derive the asymptotic behavior of the eigenvalues.

MSC:

34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
34B40 Boundary value problems on infinite intervals for ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
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References:

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