# Electrostatic equilibria on the unit circle via Jacobi polynomials

@article{Johnson2020ElectrostaticEO, title={Electrostatic equilibria on the unit circle via Jacobi polynomials}, author={Krista E Johnson and Brian Simanek}, journal={arXiv: Mathematical Physics}, year={2020} }

We use classical Jacobi polynomials to identify the equilibrium configurations of charged particles confined to the unit circle. Our main result unifies two theorems from a 1986 paper of Forrester and Rogers.

## Figures from this paper

## 2 Citations

HIGH-LOW TEMPERATURE DUALITIES FOR THE CLASSICAL β-ENSEMBLES

- Mathematics
- 2021

The loop equations for the [Formula: see text]-ensembles are conventionally solved in terms of a [Formula: see text] expansion. We observe that it is also possible to fix [Formula: see text] and…

High-low temperature dualities for the classical $\beta$-ensembles

- Mathematics
- 2021

The loop equations for the β-ensembles are conventionally solved in terms of a 1/N expansion. We observe that it is also possible to fix N and expand in inverse powers of β. At leading order, for the…

## References

SHOWING 1-10 OF 14 REFERENCES

An electrostatics model for zeros of general orthogonal polynomials

- Mathematics
- 2000

We prove that the zeros of general orthogonal polynomials, subject to certain integrability conditions on their weight functions determine the equilibrium position of movable n unit charges in an…

Electrostatics, Hyperbolic Geometry and Wandering Vectors

- Mathematics
- 2004

A family of planar discrete electrostatic systems on the unit circle with finitely atomic external fields is considered. The geometry of particles in the external field yielding a given minimum…

Electrostatics and the zeros of the classical polynomials

- Physics
- 1986

New interpretations of the zeros of the classical polynomials as the equilibrium positions of two-dimensional electrostatic problems are given. The electrostatic problems solved include determining…

A minimum energy problem and Dirichlet spaces

- Mathematics
- 2001

We analyze a minimum energy problem for a discrete electrostatic model in the complex plane and discuss some applications. A natural characteristic distinguishing the state of minimum energy from…

An Electrostatic Interpretation of the Zeros of Paraorthogonal Polynomials on the Unit Circle

- MathematicsSIAM J. Math. Anal.
- 2016

It is shown that the zeros of every paraorthogonal polynomial mark the locations of a set of particles that are in electrostatic equilibrium with respect to a particular external field.

Orthogonal Polynomials

- Mathematics
- 2005

In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed.

Convexity: An Analytic Viewpoint

- Mathematics
- 2011

Convexity is important in theoretical aspects of mathematics and also for economists and physicists. In this monograph the author provides a comprehensive insight into convex sets and functions…

‘A’

- MedicineComposites Engineering: An A–Z Guide
- 2021

A fluorescence-imaging-based endoscopic capsule that automates the detection process of colorectal cancer was designed and developed in the lab and offered great possibilities for future applicability in selective and specific detection of other fluorescently labelled cancers.