Asymptotic lattices, good labellings, and the rotation number for quantum integrable systems

Monique Dauge; Michael A. Hall; San Vũ Ngọc

doi:10.3934/dcds.2022120

Article Contents

2022, Volume 42, Issue 12: 5683-5735. Doi: 10.3934/dcds.2022120

This issue Previous Article Liouville type theorems for stable solutions of the weighted fractional Lane-Emden system Next Article Diffusing orbits along chains of cylinders

Asymptotic lattices, good labellings, and the rotation number for quantum integrable systems

1.
Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France
2.
University of Southern California, Los Angeles, USA

^*Corresponding author: San Vũ Ngọc
^*Corresponding author: San Vũ Ngọc

Received: January 2021

Revised: June 2022

Early access: September 2022

Published: December 2022

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Abstract

This article introduces the notion of good labellings for asymptotic lattices in order to study joint spectra of quantum integrable systems from the point of view of inverse spectral theory. As an application, we consider a new spectral quantity for a quantum integrable system, the quantum rotation number. In the case of two degrees of freedom, we obtain a constructive algorithm for the detection of appropriate labellings for joint eigenvalues, which we use to prove that, in the semiclassical limit, the quantum rotation number can be calculated on a joint spectrum in a robust way, and converges to the well-known classical rotation number. The general results are applied to the semitoric case where formulas become particularly natural.

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Mathematics Subject Classification: Primary: 81S10, 81Q20; Secondary: 58J40, 58J50, 65L09.

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Figure 1. The semitoric rotation number

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Figure 2. Joint spectrum of the Spherical Pendulum. Joint eigenvalues are organized along vertical lines indexed by $ j $

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Figure 3. Classical (left) and quantum (right) rotation numbers for an axisymmetric Schrödinger operator on the sphere $ \mathbb{S}^2 $ with potential $ V = z $ (quantum spherical pendulum)

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Figure 4. The labelling algorithm

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Figure 5. The point $ \tilde\mu $ is pushed onto the thickened path

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