Spherical Harmonic Oscillators

TL;DR

The paper develops a rigorous spectral theory for a non-factorizable spherical harmonic oscillator on $S^d$, defined by $L_{\omega}=\Delta_{S^d}+\omega^2 r^2$. It proves essential self-adjointness with a compact resolvent, identifies the ground-state structure, and derives explicit spectral formulas in the two-dimensional case, plus a general description for arbitrary $d$, complemented by a detailed partition-function analysis and concrete $d=1$ and $d=2$ examples. A key finding is the discontinuous dependence of ground-state properties on the dimension, with $d=2$ acting as a critical threshold, and the work hints at a possible connection to the two-dimensional principal chiral model through a decomposition into Gaussian and spherical-oscillator sectors and speculative product-form partition functions. These results provide a mathematically controlled model of spherical oscillators and yield tractable, testable predictions for heat-kernel/partition-function behavior with potential implications for 2D quantum field theory.

Abstract

A linear quantum harmonic oscillator factors into one dimensional oscillators and can be solved using creation and annihilation operators. We consider a spherical analogue. This analogue does not factor. The two dimensional case is critical, and we compute the spectrum and partition function. This is of interest because the spherical oscillator is potentially relevant to chiral models in 2d quantum field theory.

Theorems & Definitions (14)

• Definition 1
• Theorem 1.1
• Theorem 2.1
• proof
• Remark
• Theorem 2.2
• proof
• Theorem 3.1
• Remark
• proof