Dirichlet-Neumann bracketing for a class of banded Toeplitz matrices

TL;DR

This paper investigates boundary conditions for self-adjoint banded Toeplitz matrices, establishing the existence of Dirichlet-Neumann bracketing conditions for a specific class including powers of the discrete Laplacian, and providing spectral gap bounds.

Contribution

It introduces boundary conditions satisfying Dirichlet-Neumann bracketing for a class of banded Toeplitz matrices, including integer powers of the discrete Laplacian, and derives spectral gap estimates.

Findings

Existence of boundary conditions satisfying Dirichlet-Neumann bracketing for certain Toeplitz matrices.

Lower bounds on the spectral gap above the lowest eigenvalue.

Application to matrices including powers of the discrete Laplacian.

Abstract

We consider boundary conditions of self-adjoint banded Toeplitz matrices. We ask if boundary conditions exist for banded self-adjoint Toeplitz matrices which satisfy operator inequalities of Dirichlet-Neumann bracketing type. For a special class of banded Toeplitz matrices including integer powers of the discrete Laplacian we find such boundary conditions. Moreover, for this class we give a lower bound on the spectral gap above the lowest eigenvalue.