Limit absorption and Green function estimates for matrix-valued periodic operators

TL;DR

The paper establishes limit absorption principles for matrix-valued periodic lattice operators on $\ell^2(\mathbb{Z}^d,\mathbb{C}^L)$, including regimes with band-touching Weyl points. It combines Bloch–Floquet decomposition, coarea-based analysis for regular points, a parametric Morse-lemma treatment for indefinite critical points, and novel oscillatory-integral bounds to control Weyl-point contributions, assembling these local analyses into a global LAP via partitions of unity and Riesz projections. In $d=3$, the authors prove the existence and Hölder continuity of the damped resolvent limits $${\cal R}^{E\pm i0}_\alpha$$ for $\alpha>\tfrac{5}{4}$ (with refined exponents depending on the region of the Brillouin zone); they also derive detailed Green-function decay near Morse critical points and near Weyl points. The work yields a robust framework for spectral and scattering analysis of matrix-valued periodic operators in higher dimensions and lays groundwork for extensions to Schrödinger-type operators with ultraviolet cutoffs.

Abstract

The boundary value of the resolvent of a generic periodic tight-binding Hamiltonian with matrix symbols is shown to satisfy a limit absorption principle which is continuous in energy in dimensions $d=3$, and in dimension $d=2$ away from critical points of the energy bands corresponding to van Hove singularities. The analysis away from critical points of the energy bands is based on the coarea formula, while at the critical points it involves a parametric Morse lemma and stationary phase arguments. In particular, at Weyl points a new type of oscillatory integrals is dealt with.

Theorems & Definitions (27)

• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Proposition 10
• Remark 11
• Lemma 12
• Lemma 13
• Lemma 14