On global Schrödinger kernel estimate and eigenvalue problem (Q1377062)

Let \(M\) be an \(m\)-dimensional (\(m \geq 3\)) compact Riemannian manifold with boundary \(\partial M \neq 0\) and Ricci curvature bounded from below. The author considers the following Neumann eigenvalue problem: \[ \Delta u - qu = -\eta u \quad\text{in } M,\qquad {\partial u \over \partial \nu} = 0 \quad\text{on } \partial M. \] If \(\partial M\) satisfies the ``interior rolling R-ball condition'', the author proves Harnack's inequality, upper estimates for the solutions, and the following estimate for the gap between the \(k\)-th and first eigenvalues: \(\Gamma_k \geq \text{const } k^{2/m}\).