Some eigenvalue inequalities for a class of Jacobi matrices

AbstractWe consider the class of Jacobi (tridiagonal) matrices T = L+D>, where L is the negative of the discrete Laplacian and D is a diagonal matrix. We prove the inequality λ1(T) ⩾ λ1(T̃), where λ1(T) represents the lowest eigenvalue of the matrix T and where T̃ = L+D̃ with D̃ being the “symmetric-increasing rearrangement” of D. The proof follows from rearrangement inequalities going back at least to Hardy, Littlewood, and Pólya and is the one-dimensional discrete analogue of a well-known result for Schrödinger operators. We also prove that the gap, λ2 − λ1, is increased by strictly symmetric-increasing perturbations in the case that D is symmetric. Finally, we give an inequality relating the lowest eigenvalues of four Jacobi matrices of the form T= L+D when their potentials D obey certain conditions