On an estimate for the first eigenvalue of the Sturm-Liouville problem (Q2640748)

The first eigenvalue \(\lambda\) of the Sturm-Liouville problem \((P(x)y')'=\lambda y\) on \(0<x<1\); \(y(0)=0\), \(y(1)=0\) is considered when the potential P(x) is a nonnegative function for which \(\int^{1}_{0}P(x)^{\alpha}dx=1\), where \(\alpha\) is a real number. An analogous problem for the equation \(y''=\lambda P(x)y\) was discussed previously by the authors [Usp. Mat. Nauk 39, No.2, 151-152 (1984; Zbl 0548.34027)]. If \(\alpha >-1/2\) and \(\alpha\neq 0\) then sup \(\lambda\) \(=C\), where C(\(\alpha\)) is a constant, the \(\lambda\) value being any small positive value. If \(\alpha\leq -1\) then inf \(\lambda\) \(=C>0\) at any (even very large) \(\lambda\) value. If \(0<\alpha\) a formula was derived for C. Proofs are given.