Hilbert Spaces of bounded one dimensional non-linear oscillators are studied.
It is shown that the eigenvalue structure of all such oscillators have the same
general form. They are dependent only on the ground state energy of the system
and a single functional $\lambda(H)$ of the Hamiltonian $H$ whose form depends
explicitly on $H$. It is also found that the Hilbert Space of the non-linear
oscillator is unitarily inequivalent to the Hilbert Space of the simple
harmonic oscillator, providing an explicit example of Haag's Theorem. A number
operator for the nonlinear oscillator is constructed and the general form of
the partition function and average energy of an non-linear oscillator in
contact with a heat bath is determined. Connection with the WKB result in the
semi-classical limit is made. This analysis is then applied to the specific
case of the $x^4$ anharmonic oscillator.