On sharp lower bound of the gap for the first two eigenvalues in the Schrödinger operator (Q1949021)

Let \(\Omega\) (\(\subset \mathbb{R}^n\)) be a smooth strictly convex bounded domain with diameter \(d_\Omega\), \(W\) a nonnegative convex potential on \(\overline\Omega\) et \(0<\lambda_1<\lambda_2\) the first two Dirichlet eigenvalues for the Schrödinger operator \(\Delta+W\). \textit{I. M. Singer} et al., [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 12, 319--333 (1985; Zbl 0603.35070)] established the estimate \(\lambda_2-\lambda_1\geq \pi^2/4d_\Omega^2\), which has been improved by \textit{Q. Yu} and \textit{J.-Q. Zhong} [Trans. Am. Math. Soc. 294, 341--349 (1986; Zbl 0593.53030)] who proved \(\lambda_2-\lambda_1\geq\pi^2/d_\Omega^2\). In this paper, He gives a variant for the proof of the last authors.