Quantum Mechanics Model on K\"ahler conifold

TL;DR

This paper introduces an exactly-solvable quantum oscillator model on K"ahler conifolds, revealing unique spectral properties and quantum corrections, and connects it to MIC-Kepler systems with potential fractional spin interpretations.

Contribution

It presents a novel exactly-solvable quantum oscillator model on K"ahler spaces with conic singularities, including quantum corrections and a transformation to MIC-Kepler-like systems.

Findings

Energy spectrum is nondegenerate for non-constant curvature spaces.

Quantum corrections affect energy and coupling constants.

Exact spectrum of the generalized MIC-Kepler system is obtained.

Abstract

We propose an exactly-solvable model of the quantum oscillator on the class of K\"ahler spaces (with conic singularities), connected with two-dimensional complex projective spaces. Its energy spectrum is nondegenerate in the orbital quantum number, when the space has non-constant curvature. We reduce the model to a three-dimensional system interacting with the Dirac monopole. Owing to noncommutativity of the reduction and quantization procedures, the Hamiltonian of the reduced system gets non-trivial quantum corrections. We transform the reduced system into a MIC-Kepler-like one and find that quantum corrections arise only in its energy and coupling constant. We present the exact spectrum of the generalized MIC-Kepler system. The one-(complex) dimensional analog of the suggested model is formulated on the Riemann surface over the complex projective plane and could be interpreted as a system with fractional spin.