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On the distribution of the eigenvalues of a class of indefinite eigenvalue problems. (English) Zbl 0736.34071
The authors consider boundary eigenvalue problems of the form $$y\sp{(n)}(x)+\sum\sp n\sb{\nu=2}f\sb \nu(x)y\sp{(n-\nu)}(x)=\lambda r(x)y(x)\qquad (x\in[0,1]),$$ $$U\sb{\nu0}(y)+U\sb{\nu1}(y)=0\qquad (\nu=1,2,\dots,n),$$ where $n\ge 2$, the coefficients $f\sb \nu$ belong to $L\sb 1(0,1)$, $r:[0,1]\to\bbfR\backslash\{0\}$ is a step function, and the two-point boundary conditions are assumed to be normalized. For such indefinite problems the authors introduce the notion of regularity which, in a natural way, generalizes that of Birkhoff regularity concerning definite problems. For such regular boundary value problems the authors determine the asymptotic distribution of the eigenvalues. For special problems concerning second order differential equations $(n=2)$ similar results have been proved by {\it R. E. Langer} [Trans. Am. Math. Soc. 31, 1-24 (1929)] and {\it A. B. Mingarelli} [Lect. Notes Math. 1032, 375-383 (1983; Zbl 0563.34022)].

34L10Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions (ODE)
34L20Asymptotic distribution of eigenvalues for OD operators