The General Structure of Eigenvalues of Nonlinear Oscillators
Abstract
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 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 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 an nonlinear oscillator in contact with a heat bath is determined. Connection with the WKB result in the semiclassical limit is made. This analysis is then applied to the specific case of the $x^4$ anharmonic oscillator.
 Publication:

arXiv eprints
 Pub Date:
 November 1996
 DOI:
 10.48550/arXiv.physics/9611006
 arXiv:
 arXiv:physics/9611006
 Bibcode:
 1996physics..11006S
 Keywords:

 Mathematical Physics;
 High Energy Physics  Theory;
 Quantum Physics
 EPrint:
 25 pages, written in RevTex, no figures. This paper has been substantially rewritten with new results added