GAP estimates of Schrödinger operator (Q1365140)

The aim of this paper is to give a lower bound for the gap \(\Gamma_k:= (\eta_k-\eta_1)\) of \(k\)th eigenvalues \(\eta_k\) and \(\eta_1\) of the following Neumann eigenvalue problem \[ \Delta u-qu= -\eta u \quad\text{in }M \qquad\text{and}\quad \frac{\partial u}{\partial\nu} =0\quad\text{on }\partial M. \] Here \(M\) is an \(m\)-dimensional compact Riemannian manifold \((m\geq 3)\) and \(\partial M\neq\emptyset\) the boundary on \(M\) and \(q\in C^2(M)\). More precisely, the author proves, under some geometrical assumptions on \(M\) and convexity conditions for \(\partial M\), that the gap \(\Gamma_k\) satisfies \(\Gamma_k\geq \alpha_1 k^{2/m}\), for all \(k=2,3,\dots\) and some constant \(\alpha_1\) depending on the geometrical quantities of \(M\), \(m\), and the oscillation of \(q\) over \(M\). The techniques used are global estimates for the first eigenfunction and similar arguments as used by \textit{S.-Y. Cheng} and \textit{P. Li} [Comment. Math. Helv. 56, 327-338 (1981; Zbl 0484.53034)].