Upper and lower bounds for the eigenvalues of three types of matrices M are established. If M is written as the sum of a diagonal matrix D plus a matrix A, the real parts of the eigenvalues of M must ...

We prove the optimal lower bound $\lambda_2 - \lambda_1 \geq 3\pi^2/d^2$ for the difference of the first two eigenvalues of a one-dimensional Schrödinger operator ...