Suppose that there is an object (i.e. a wavefunction) containing $N+M$ particles, say, a person holding a mobile phone, which may or may not interact each other. Suppose also that the object needs to be split into an $N$-particle wavefunction for the person and the $M$-particle wavefunction for the phone. Intuitively the successor of the $N+M$-particle wavefunction should be the $N$-particle wavefunction, but the interacting-particle version of the Schrödinger equation does not seem to interpret it as the successor. Absorbing the $M$-particle wavefunction into the potential function does not really seem to help either.
I'm asking for a known mathematical solution (or absence of such thereafter) of this apparent paradox.
P.S. I'm a freshman in college and haven't learnt enough math to prove these sort of things myself. This is not an assignment either, as it's vacation time for most of the colleges here; while K-12 started their classes (even online ones) a month later than usual, most colleges started their lectures only two weeks later than usual.