Uniform Sobolev inequalities and absolute continuity of periodic operators

Abstract:

We establish certain uniform $L^{p}-L^{q}$ inequalities for a family of second order elliptic operators of the form $( {\mathbf {D}} + {\mathbf {k}} ) A ( {\mathbf {D}}+ {\mathbf {k} })^{T}$ on the $d$-torus, where ${\mathbf {D}} =-i\nabla , {\mathbf {k}}\in {\mathbb {C}} ^{d}$ and $A$ is a symmetric, positive definite $d\times d$ matrix with real constant entries. Using these Sobolev type inequalities, we obtain the absolute continuity of the spectrum of the periodic Dirac operator on ${\mathbb R}^{d}$ with singular potential. The absolute continuity of the elliptic operator div$(\omega ( {\mathbf {x}})\nabla )$ on ${\mathbb R}^{d}$ with a positive periodic scalar function $\omega ( {\mathbf {x}} )$ is also studied.