The general structure of eigenvalues in nonlinear oscillators

Hilbert spaces of bounded one-dimensional nonlinear oscillators are studied. It is shown that the eigenvalue structure of all such oscillators have the same general form. They depend only on the ground state energy of the system and a single function λ(H) of the Hamiltonian operator H. It is also found that the Hilbert space of the nonlinear 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 a nonlinear oscillator in contact with a heat bath is determined. Connection with the WKB result in the semiclassical limit is made. The analysis is then applied to the case of the cursive Greek chi4 anharmonic oscillator as an explicit example.