  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.
