Root Asymptotics for the Eigenfunctions of Univariate Differential Operators

On asymptotics of polynomial eigenfunctions for exactly solvable differential operators

In this paper we study the class of differential operators T = Pk j=1 QjD j with polynomial coefficients Qj in one complex variable satisfying the condition degQj ≤ j with equality for at least one j. We show that if degQk < k then the root with the largest modulus of the nth degree eigenpolynomial pn of T tends to infinity when n → ∞, as opposed to the case when degQk = k, which we have treate...

Monomiality Principle and Eigenfunctions of Differential Operators

On Polynomial Eigenfunctions for a Class of Differential Operators

Jacobi polynomials are solutions of the differential equation (z − 1)f ′′(z) + (az + b)f ′(z) + cf(z) = 0, (1) where a, b, c are constants satisfying a > b, a + b > 0 and c = n(1 − a − n) for some nonnegative integer n. It is a classical fact that the zeros of the Jacobi polynomials lie in the interval [−1, 1], and that their density in this interval is proportional to 1/ √ 1 − |z|2 in the limi...

Asymptotics of polynomials and eigenfunctions

We review some recent results on asymptotic properties of polynomials of large degree, of general holomorphic sections of high powers of positive line bundles over Kähler manifolds, and of Laplace eigenfunctions of large eigenvalue on compact Riemannian manifolds. We describe statistical patterns in the zeros, critical points and Lp norms of random polynomials and holomorphic sections, and the ...

Homogenized Spectral Problems for Exactly Solvable Operators: Asymptotics of Polynomial Eigenfunctions

Consider a homogenized spectral pencil of exactly solvable linear differential operators Tλ = Pk i=0 Qi(z)λ k−i d i dzi , where each Qi(z) is a polynomial of degree at most i and λ is the spectral parameter. We show that under mild nondegeneracy assumptions for all sufficiently large positive integers n there exist exactly k distinct values λn,j , 1 ≤ j ≤ k, of the spectral parameter λ such tha...