1996/11/07 by Achilles D. Speliotopoulos, Speliotopoulos, Achilles D.
Mathematics · Physics and Astronomy · #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Mechanical and Optical Resonators #Quantum Electrodynamics and Casimir Effect #Quantum Mechanics and Non-Hermitian Physics #Quantum Physics (quant-ph) #hep-th #math-ph #math.MP #quant-ph
paper · pdf · doi:10.48550/arxiv.physics/9611006
25 pages, written in RevTex, no figures. This paper has been substantially re-written with new results added
openalex publication_date 1996/11/07 · arxiv created 1998/05/25 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 λ(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 x4 anharmonic oscillator.