Feynman formulas for qp- and pq-quantization of some Vladimirov type time-dependent Hamiltonians on finite adeles

Abstract

Let Q be the d-dimensional space of finite adeles over the algebraic number field K and let P=Q∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P=Q^*$$\\end{document} be its dual space. For a certain type of Vladimirov type time-dependent Hamiltonian HV(t):Q×P→C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_V(t):Q\ imes P\\rightarrow {\\mathbb {C}}$$\\end{document} we construct the Feynman formulas for the solution of the Cauchy problem with the Schrödinger operator where the caret operator stands for the qp- or pq-quantization.