Valeurs propres d'une classe d'opérateurs elliptiques dégénérés. (Eigenvalues of a class of degenerate elliptic operators) (Q760606)

A special type of degenerated elliptic operators is considered, represented by \(L:=\partial (x^ h\partial /\partial x)/\partial x+x^{\alpha}\partial^ 2/\partial y^ 2\) \((-\infty <h<2,\alpha >2)\) in rectangular finite or infinite domains - more accurately speaking: the selfadjoint extension of L in a suitable Hilbert space. The aim is to derive precisely the asymptotic behavior of the eigenvalues. Let N(\(\lambda)\) denote the number of eigenvalues not exceeding \(\lambda\). The authors intend to give optimal estimates of the rest of N(\(\lambda)\), \(\lambda\) \(\to \infty\), in relation to \(\alpha\), h. By separation of variables the problem is reduced on the treatment of the ordinary differential operator \(P:=d(t^ hd/dt)/dt-t^{\alpha}\) \((- \infty <h<2\), \(\alpha >2)\) on a half-line. Then \textit{E. C. Titchmarsh}'s classical method [Eigenfunction expansions associated with second-order differential equations. Vol. I (1946; Zbl 0061.135) and Vol. II (1958; Zbl 0097.276)] is essentially relied on. For similar but more general results, received with different methods, see \textit{P. Bolley, J. Camus} and \textit{Pham The Lai}, Lect. Notes Math. 660, 33-46 (1978; Zbl 0389.35036) and \textit{Pham The Lai} and \textit{D. Robert} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 6, 335-366 (1979; Zbl 0422.35062)].