The General Structure of Eigenvalues of Non-linear Oscillators

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.