Mathematical computability questions for some classes of linear and non-linear differential equations originated from Hilbert's tenth problem

Abstract

Inspired by Quantum Mechanics, we reformulate Hilbert's tenth problem in the domain of integer arithmetics into problems involving either a set of infinitely-coupled non-linear differential equations or a class of linear Schr\"odinger equations with some appropriate time-dependent Hamiltonians. We then raise the questions whether these two classes of differential equations are computable or not in some computation models of computable analysis. These are non-trivial and important questions given that: (i) not all computation models of computable analysis are equivalent, unlike the case with classical recursion theory; (ii) and not all models necessarily and inevitably reduce computability of real functions to discrete computations on Turing machines. However unlikely the positive answers to our computability questions, their existence should deserve special attention and be satisfactorily settled since such positive answers may also have interesting logical consequence back in the classical recursion theory for the Church-Turing thesis.