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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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##### References:
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