On some eigenvalue problems (Q2752836)

Let \((M,g)\) be a compact Riemannian manifold of dimension \(m\geq 3\) with smooth non-empty boundary. Let \(\Delta\) be the scalar Laplacian. If \(q\) is a non-negative \(C^2\) function on \(M\), let NEWLINE\[NEWLINE\text{spec}(M,q)= \{\eta_1\leq\eta_2\leq\dots\}NEWLINE\]NEWLINE be the spectrum of the Schrödinger operator \(\Delta-q\) with Neumann boundary conditions. NEWLINENEWLINENEWLINEThe author describes a computable lower estimate for \(\eta_k\) when \(q=0\) and for \(\Gamma_k:=\eta_k-\eta_1\) when \(q\neq 0\) in terms of geometrical quantities of \(M\), \(\partial M\), \(m\), and the \(C^2\) norm of \(q\). The proof uses heat kernel estimates and methods of \textit{S. Y. Cheng} and \textit{P. Li} [Comment. Math. Helv. 56, 327-338 (1981; Zbl 0484.53034)] and \textit{J. Wang} [Pac. J. Math. 178, No. 2, 377-398 (1997; Zbl 0882.35018)]; see also previous work by the author [Pac. J. Math. 178, No. 2, 225-240 (1997; Zbl 0886.35102) and Math. Z. 227, No. 1, 69-81 (1998; Zbl 0894.35077)].NEWLINENEWLINEFor the entire collection see [Zbl 0953.00035].