Boundedness of Operators Related to a Degenerate Schrödinger Semigroup

In this work we search for boundedness results for operators related to the semigroup generated by the degenerate Schrödinger operator Lu=−1ωdivA⋅∇u+Vu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${{\mathscr{L}}} u = -\frac {1}{\omega } \text {div} A\cdot \nabla u +V u$\end{document}, where ω is a weight, A is a matrix depending on x and satisfying λω(x)|ξ|2 ≤ A(x)ξ ⋅ ξ ≤Λω(x)|ξ|2 for some positive constants λ, Λ and all x, ξ in ℝd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}^{d}$\end{document}, assuming further suitable properties on the weight ω and on the non-negative potential V. In particular, we analyze the behaviour of T∗, the maximal semigroup operator, L−α/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${{\mathscr{L}}}^{-\alpha /2}$\end{document}, the negative powers of L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${{\mathscr{L}}}$\end{document}, and the mixed operators L−α/2Vσ/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${{\mathscr{L}}}^{-\alpha /2}V^{\sigma /2}$\end{document} with 0 < σ ≤ α on appropriate functions spaces measuring size and regularity. As in the non degenerate case, i.e. ω ≡ 1, we achieve these results by first studying the case V = 0, obtaining also some boundedness properties in this context that we believe are new.