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The support of the equilibrium measure in the presence of a monomial external field on \([-1,1]\). (English) Zbl 0943.31001
Corresponding to a continuous function \(Q:[-1,1]\to \mathbb{R}\), there is a unique Borel probability measure \(\mu\) on \([-1,1]\), called the equilibrium measure, such that, for some constant \(F\), we have \(U^\mu+ Q\geq F\) on \([-1,1]\) with equality on \(\operatorname {supp} \mu\); here \(U^\mu(x)= -\int\log|x-t|d\mu(t)\). The authors study the support \(S\) of \(\mu\) in the case where \(Q(x)= -cx^{2m+1}\) with \(m\in \mathbb{N}\) and \(c>0\). Their main result asserts that corresponding to each \(m\) there are three critical values \(c_j\) with \(0< c_1< c_2< c_3\) such that (i) for \(0< c\leq c_1\), we have \(S= [-1,1]\), (ii) for \(c_1< c\leq c_2\), we have \(S= [a,1]\), where \(-1< a< 0\), (iii) for \(c_2< c< c_3\), we have \(S= [a_1,b_1]\cup [a_2,1]\), where \(-1< a_1< b_1< a_2< 1\), (iv) for \(c_3\leq c\), we have \(S= [a,1]\), where \(0< a< 1\). The numbers \(a\), \(a_1\), \(a_2\), \(b_1\) depend on \(c\). This result answers a question of P. Deift, T. Kriecherbauer and K. T.-R. McLaughlin [J. Approx. Theory 95, 388-475 (1998; Zbl 0918.31001)].

MSC:
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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