CORRIGENDUM: “HOMOGENIZED SPECTRAL PROBLEMS FOR EXACTLY SOLVABLE OPERATORS: ASYMPTOTICS OF POLYNOMIAL EIGENFUNCTIONS” JULIUS BORCEA † , RIKARD BØGVAD, AND BORIS SHAPIRO * Abstract. Here we provide a correct proof of Proposition 6 of [2]. No other results of the latter paper are affected. 1. Necessary results and corrected proof To make this note self-contained we briefly recall the basic set-up of [2]. Given a (k + 1)-tuple of polynomials (Q k (z),Q k−1 (z), ..., Q 0 (z)) with deg Q i (z) ≤ i consider the homogenized spectral pencil of differential operators given by: T λ = k  i=0 Q i (z)λ k−i d i dz i . (1.1) Introduce the algebraic curve Γ associated with T λ and given by the equation k  i=0 Q i (z)w i =0, (1.2) where the polynomials Q i (z)= ∑ i j=0 a i,j z j are the same as in (1.1). The curve Γ and its associated pencil T λ are called of general type if the following two nondegeneracy requirements are satisfied: (i) deg Q k (z)= k (i.e., a k,k = 0), (ii) no two roots of the (characteristic) equation a k,k + a k−1,k−1 t + ... + a 0,0 t k =0 (1.3) lie on a line through the origin (in particular, 0 is not a root of (1.3)). The first statement of [2] we need is as follows. Proposition 1. If the characteristic equation (1.3) has k distinct solutions α 1 ,α 2 , ...,α k and satisfies the preceding nondegeneracy assumptions (in particular, these imply that a 0,0 =0 and a k,k =0) then (i) for all sufficiently large n there exist exactly k distinct eigenvalues λ n,j , j =1,...,k, such that the associated spectral pencil T λ has a polynomial eigenfunction p n,j (z) of degree exactly n, (ii) the eigenvalues λ n,j split into k distinct families labeled by the roots of (1.3) such that the eigenvalues in the j -th family satisfy lim n→∞ λ n,j n = α j , j =1,...,k. 2000 Mathematics Subject Classification. 30C15, 31A35, 34E05. Key words and phrases. Asymptotic root-counting measure, Cauchy transform, homogenized spectral problem, exactly solvable operator. † J. B. unexpectedly passed away on April 8, 2009 at the age of 40. * Corresponding author. 1