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An analog of a theorem of Keldysh for a multipoint problem of de la Vallée Poussin. (English. Russian original) Zbl 0789.34023
Russ. Acad. Sci., Dokl., Math. 46, No. 2, 307-310 (1993); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 326, No. 4, 587-591 (1992).
The authors study the problem (1) \(x^{(n)} +p_ 1(t)x^{(n-1)} +\cdots+ p_ n(t) x=\lambda x\), \(a\leq t\leq b\), (2) \(x^{(i)} (a_ k)=0\), \(i=0,\dots,\nu_ k-1\), \(k=1,\dots,N\), \(\nu_ 1+ \cdots+\nu_ N=n\), where \(a=a_ 1<a_ 2<\cdots<a_ N=b\). They establish a representation of the residue of the Green function of problem (1), (2) in a neighborhood of the eigenvalue \(\lambda_ 0\). Both a simple eigenvalue and the case of multiplicity are considered.
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators