On external fields created by fixed charges

Orive, R.; Lara, Joaquín F. Sánchez

doi:10.1016/j.jmaa.2018.03.078

Cited by 4 publications

(11 citation statements)

References 29 publications

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“…, q), then η T is a positive measure on the whole real axis. From the characterization of the equilibrium measure given in Proposition 4.8 and the identity (2.1), we may thus deduce that µ T = η T is supported on the whole axis R. This was the case considered in the previous [15].…”

Section: Weak External Field Created By a Finite Number Of Charges: Asupporting

confidence: 51%

“…As in [11] and [14]- [15], we proceed by studying the evolution of the equilibrium measure µ t in the external field Q in (3.2) as the mass t ∈ (0, T ) approaches the limit value T . In the next theorem, we show that the measures µ t have a weak-* limit as t tends to T , namely the equilibrium measure µ T , solution of the weakly admissible equilibrium problem.…”

Section: Equilibrium Measures On the Real Linementioning

confidence: 99%

“…It describes the dynamics of the equilibrium measure µ T in the weakly admissible external field Q as the parameter γ grows from 0 to 1. In different situations, such as those studied in the papers [11,14,15], the measure µ T , limit of the equilibrium measures µ t when t → T , always had its support equal to the whole real axis. As the following theorem shows, the behavior of µ T in the present situation can be different.…”

Section: )mentioning

confidence: 99%

“…The particular case where Q has an additional polynomial part of even degree was considered in [14]. Then, in [15], the case of rational external fields of type (1.1) (without polynomial part) was treated when all γ k > 0 (and, thus, all z k ∈ C \ R). One of the contributions of this paper is to deal with both positive and negative charges (that is, a mixture of attractors and repellents), where in principle, for γ k < 0, z k ∈ R is allowed.…”

Section: Introductionmentioning

confidence: 99%

“…One of the contributions of this paper is to deal with both positive and negative charges (that is, a mixture of attractors and repellents), where in principle, for γ k < 0, z k ∈ R is allowed. Though, in general, the study of weighted equilibrium problems on a conductor Σ deals with probabilistic equilibrium measures µ Q with µ Q (Σ) = 1, the authors of [11], [14] and [15] have preferred to consider equilibrium measures with varying mass µ t = µ t,Q , in the sense that µ t (Σ) = t > 0. This is similar to the dynamical approach proposed in the seminal work by V. Buyarov and E. A. Rakhmanov [4].…”

Section: Introductionmentioning

confidence: 99%

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Equilibrium problems in weakly admissible external fields created by pointwise charges

Orive

Lara

Wielonsky

2019

Journal of Approximation Theory

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The main subject of this paper is equilibrium problems on an unbounded conductor Σ of the real axis in the presence of a weakly admissible external field. An admissible external field Q on Σ satisfies, along with other mild conditions, the following growth property at infinity:This condition guarantees the existence and uniqueness of the equilibrium measure in the presence of Q, and the compactness of its support. In the last 10-15 years, several papers have dealt with weakly admissible external fields, in the sense that Q satisfies a weaker condition at infinity, namely,Under this last assumption, there still exists a unique equilibrium measure in the external field Q, but the support need not be a compact subset of Σ anymore.In most examples considered in the literature the support is indeed unbounded. Our main goal in this paper is to illustrate this topic by means of a simple class of external fields on the real axis created by a pair of attractive and repellent charges in the complex plane, and to study the dynamics of the associated equilibrium measures as the strength of the charges evolves. As one of our findings, we exhibit configurations where the support of the equilibrium measure in a weakly admissible external field is a compact subset of the real axis. To achieve our goal, we extend some results from potential theory, known for admissible external fields, to the weakly admissible case. These new results may be of independent interest. Finally, the so-called signed equilibrium measure is an important tool in our analysis. Its relationship with the (positive) equilibrium measure is also explored.

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Section: Weak External Field Created By a Finite Number Of Charges: Asupporting

confidence: 51%

Section: Equilibrium Measures On the Real Linementioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Equilibrium problems in weakly admissible external fields created by pointwise charges

Orive

Lara

Wielonsky

2019

Journal of Approximation Theory

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Weighted Fekete Points on the Real Line and the Unit Circle

Dubickas

Pritsker

2020

Comput. Methods Funct. Theory

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Weighted Fekete points are defined as those that maximize the weighted version of the Vandermonde determinant over a fixed set. They can also be viewed as the equilibrium distribution of the unit discrete charges in an external electrostatic field. While these points have many applications, they are very difficult to find explicitly, and are only known in a few (unweighted) classical cases. We give two rare explicit examples of weighted Fekete points. The first one is for the weights w(x) = |x − ai| −s on the real line, with s ≥ 1 and a = 0, while the second is for the weights w(z) = 1/|z − b| on the unit circle, with b ∈ R, b = ±1. In both cases, we provide solutions of the continuous energy problems with external fields that express the limit versions of considered weighted Fekete points problems.2010 Mathematics Subject Classification. 30C10, 30C15, 31C20, 12D10.

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