[Paper Review] Integrable and superintegrable systems of cylindrical type in magnetic fields

Ondřej Kubů|arXiv (Cornell University)|Oct 4, 2022

Quantum Mechanics and Non-Hermitian Physics4 citations

TL;DR

This master's thesis classifies integrable and superintegrable systems of cylindrical type in magnetic fields by deriving and solving quantum mechanical determining equations for second-order integrals of motion in cylindrical coordinates. It identifies all known first-order superintegrable systems and discovers numerous new classical systems with additional integrals of the form $L^2 + \ldots$ or $L_x p_y - L_y p_x + \ldots$, all of which admit $L_z$ as a first integral and allow separation of variables in cylindrical coordinates, with trajectories analyzed numerically where analytical solutions are unattainable.

ABSTRACT

The goal of this thesis is the search for integrable and superintegrable systems with magnetic field. We formulate the quantum mechanical determining equations for second order integrals of motion in the cylindrical coordinates and we find all quadratically integrable systems of the cylindrical type. Among them we search for systems admitting additional integrals of motion. We find all systems with an additional first order integral both in classical and quantum mechanics. It turns out that all these systems have already been known and no other exist. We also find all systems with an additional integral of type $L^2+\ldots$, respectively $L_y p_y-L_x p_y+\ldots$, of which the majority is new to the literature. All found superintegrable systems admit the first order integral $L_z$ and we solve their Hamilton-Jacobi and Schrödinger equations by separation of variables in the cylindrical coordinates, for the first order systems in the Cartesian coordinates as well.

Motivation & Objective

• To systematically classify integrable and superintegrable systems of cylindrical type under magnetic fields.
• To derive and solve quantum mechanical determining equations for second-order integrals of motion in cylindrical coordinates.
• To identify all systems admitting additional first-order and second-order integrals beyond the standard quadratic integrability.
• To analyze the separability of Hamilton-Jacobi and Schrödinger equations in cylindrical and Cartesian coordinates.
• To explore classical-quantum correspondence and identify candidates for higher-order integrals in systems with bounded, closed trajectories.

Proposed method

• Formulation of quantum mechanical determining equations for second-order integrals of motion in cylindrical coordinates.
• Solution of the determining equations under constraints on the magnetic field and scalar potential, including $\psi(\phi)$-dependent terms and $\hbar^2$-corrections.
• Use of gauge-invariant formulations to simplify the analysis, particularly by setting $X_2 = p_z$ and $\tilde{X}_1 = p_\phi$ in suitable gauges.
• Application of physically motivated ansatzes for second-order integrals: $L^2 + \ldots$ and $L_x p_y - L_y p_x + \ldots$, leading to nonlinear ODEs for $\beta(\phi)$.
• Separation of variables in Hamilton-Jacobi and Schrödinger equations for all found systems, with solutions expressed in terms of special functions where applicable.
• Numerical analysis of trajectories for systems where analytical integration fails, focusing on boundedness and closure to identify candidates for higher-order integrals.

Experimental results

Research questions

• RQ1 Which integrable systems of cylindrical type admit additional first-order integrals in the presence of a magnetic field, and are they all already known in the literature?
• RQ2 What are the complete sets of classical systems with second-order integrals of the form $L^2 + \ldots$ or $L_x p_y - L_y p_x + \ldots$ in cylindrical coordinates under magnetic fields?
• RQ3 Do all found superintegrable systems allow separation of variables in cylindrical coordinates, and can they also be separated in Cartesian coordinates when applicable?
• RQ4 Which systems exhibit bounded, closed trajectories, and are they promising candidates for higher-order integrals beyond quadratic ones?
• RQ5 How do quantum corrections affect the scalar potential and integrals in the quantum version of these systems, and what is the nature of the classical-quantum correspondence?

Key findings

• All systems with an additional first-order integral in both classical and quantum mechanics were already known in the literature, and no new such systems were found.
• A large number of new classical systems with second-order integrals of the form $L^2 + \ldots$ or $L_x p_y - L_y p_x + \ldots$ were discovered, most of which had not been published previously.
• All found superintegrable systems admit the first-order integral $L_z$, which enables separation of variables in cylindrical coordinates.
• The Hamilton-Jacobi and Schrödinger equations for all systems were solved via separation of variables in cylindrical coordinates, with some also separable in Cartesian coordinates.
• Numerical analysis revealed that only one system (system II in Subsection 2.2.2) exhibits bounded, closed trajectories, making it a prime candidate for higher-order integrals.
• Quantum corrections to the scalar potential were found to be non-trivial for second-order integrals, and their full quantum analysis is deferred to future work.