For a time-dependent Hamiltonian function \(H(t)=H(t,\cdot ,\cdot )\) , \(t\in [0,T]\) , on the adelic phase space \(Q\times P={\mathbb {A}}_K^d\times {\mathbb {A}}_K^d\) , where \({\mathbb {A}}_K\) is the ring of finite adeles over the algebraic number field K, we consider the Schrödinger equation with \(\psi (s,q)=\psi _s(q)\) , \(q\in Q\) , \(t\in [s,T]\) , where \(\widehat{H(t)}\) is the qp-quantized Hamiltonian, which is a pseudo-differential operator on \(L^2({\mathbb {A}}_K^d)\) with symbol H(t). We show that if the Hamiltonian H(t) satisfies certain technical assumptions then there exists the solution \(\psi \) of the Schrödinger equation and can be expressed in the form of the Dyson-type series.
