Question

Is there a proof that the number of eigenstates is countable for a bound system?

Answer 1

yes there is a proof that number of eigen states is countable for a bound system but hence it is also not been proven up to till date by the scientist.
by applying lot of theorems of particular research and that is needed to be proven to study.

Answer 2

When you solve Schrödinger equation for a free particle with no boundary conditions your eigen states are indexed by quantum number k∈R and R isn't countable but if you add a harmonic oscillator potential, your eigen states are indexed by harmonic number n∈N which is countable.

More interestingly for the hydrogen atom, your potential is finite as it goes to ∞. So for E>0 you have an uncountable number of states but for E<0 there are only a countable number of eigenstates/values.

I can't think of any counter example but I have no idea how to go about proving the number of eigenstates for a general bound system is countable.

I find this interesting because if you ignore your ability to measure position and momentum it would appear that you can change the dimensionality of your system by changing your potential.