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

Let Q be the d-dimensional space of finite adeles over the algebraic number field K and let \(P=Q^*\) $P=Q^{\ast }$ be its dual space. For a certain type of Vladimirov type time-dependent Hamiltonian \(H_V(t):Q\times P\rightarrow {\mathbb {C}}\) $H_V(t):Q\times P\to \mathbb{C}$ 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.