A unified approach to universal inequalities for eigenvalues of elliptic operators
The authors present universal inequalities for the discrete spectrum of a self-adjoint operator. The main result is as follows. Let \(\mathcal{H}\) be a complex Hilbert space with inner product \(\langle\cdot,\cdot\rangle\), \(A:\mathcal{H}\supset\mathcal{D}\to\mathcal{H}\) a self-adjoint operator defined on a dense domain \(\mathcal{D}\) that is bounded from below and has a discrete spectrum \(\lambda_{1}\leq\lambda_{2}\leq\lambda_{3}\leq\cdots\), \(\{B_{k}:A(\mathcal{D})\to\mathcal{H}\}^{N}_{k=1}\) a collection of symmetric operators leaving \(\mathcal{D}\) invariant. Let \(\{u_{i}\}^{\infty}_{i=1}\) be an orthonormal system of eigenvectors of \(A\), \(u_{i}\) corresponding to \(\lambda_{i}\). Put \(\rho_{i}=\sum^{N}_{k=1}\langle [A,B_{k}]u_{i}, B_{k}u_{i}\rangle\) and \(\Lambda_{i}=\sum^{N}_{k=1}\| [A,B_{k}]u_{i}\|^{2}\). The authors prove that \[ \begin{aligned} \sum^{m}_{i=1}\rho_{i}&\leq \frac{\sum^{m}_{i=1}\Lambda_{i}}{\lambda_{m+1}-\lambda_{m}},\\ \sum^{m}_{i=1}\rho_{i}&\leq \sum^{m}_{i=1}\frac{\Lambda_{i}}{\lambda_{m+1}-\lambda_{i}},\\ \sum^{m}_{i=1}(\lambda_{m+1}-\lambda_{i})^{2}\rho_{i}&\leq \sum^{m}_{i=1}(\lambda_{m+1}-\lambda_{i})\Lambda_{i}.\end{aligned} \] The proof is based on commutator algebra and the Rayleigh-Ritz principle. The inequalities of Payne-Pólya-Weinberger, Hile-Protter, and H. C. Yang follow from the above result.
universal inequalities for the discrete spectrum of a self-adjoint operator
scientific article; zbMATH DE number 2164511