Abstract
We 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 λ 1( T ) ⩾ λ 1( T ̃ ) , where λ 1( T ) represents the lowest eigenvalue of the matrix T and where T ̃ = L + D ̃ 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, λ 2 − λ 1, 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.
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