Homogenized Spectral Problems for exactly solvable operators:Asymptotics of polynomial eigenfunctions
2009 (English)In: Publications of the Research Institute for Mathematical Sciences, ISSN 0034-5318, Vol. 45, 525-568 p.Article in journal (Refereed) Published
Consider a homogenized spectral pencil of exactly solvable linear differential operators T-lambda = Sigma(k)(i=0) Q(i)(z)lambda(k-i) d(i)/dz(i), where each Q(i)(z) is a polynomial of degree at most i and lambda is the spectral parameter. We show that under mild nondegeneracy assumptions for all sufficiently large positive integers n there exist exactly k distinct values lambda(n,j), 1 <= j <= k, of the spectral parameter lambda such that the operator T-lambda has a polynomial eigenfunction p(n,j)(z) of degree n. These eigenfunctions split into k different families according to the asymptotic behavior of their eigenvalues. We conjecture and prove sequential versions of three fundamental properties: the limits Psi(j)(Z) = lim(n ->infinity) P'(n,j) (z)/lambda(n,j)p(n,j)(z) exist, are analytic and satisfy the algebraic equation Sigma(k)(i=0)Q(i)(z)Psi(i)(j)(z) = 0 almost everywhere in CP1. As a consequence we obtain a class of algebraic functions possessing a branch near infinity is an element of CP1 which is representable as the Cauchy transform of a compactly supported probability measure.
Place, publisher, year, edition, pages
Kyoto,Japan, 2009. Vol. 45, 525-568 p.
differential equations, subharmonic functions
IdentifiersURN: urn:nbn:se:su:diva-33667DOI: 10.2977/prims/1241553129ISI: 000270167800008OAI: oai:DiVA.org:su-33667DiVA: diva2:283352