Applications Concerning Orthogonal Polynomials

The analysis of the asymptotic behavior of orthogonal polynomials was one of the driving forces for the resurgence of interest in potentials with external fields. The relationship between the two subjects can be seen as follows: consider, for example, orthogonal polynomials with respect to the so-called Freud weights W(x) = exp(-|x|λ) =: exp(-Q(x)). On applying the substitution x → n1/λx and the defining properties of orthogonal polynomials one arrives at monic polynomials P _n minimizing the integral % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfKttLearuavP1wzZbItLDhis9wBH5garm % Wu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9 % q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir % -Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGa % aeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbmaapeaapaqaa8qadaqada % WdaeaapeWaaqWaa8aabaWdbiabdcfaq9aadaWgaaWcbaWdbiabd6ga % UbWdaeqaaaGcpeGaay5bSlaawIa7aiabdwgaL9aadaahaaWcbeqaa8 % qacqGHsislcqWGUbGBcqWGrbquaaaakiaawIcacaGLPaaapaWaaWba % aSqabeaapeGaeGOmaidaaaqabeqaniabgUIiYdaaaa!3FED! $$ \int {{{{\left( {\left| {{{P}_{n}}} \right|{{e}^{{ - nQ}}}} \right)}}^{2}}} $$.