Zeroes of Wronskians of Hermite polynomials and Young diagrams

Abstract

For a certain class of partitions, a simple qualitative relation is observed between the shape of the Young diagram and the pattern of zeroes of the Wronskian of the corresponding Hermite polynomials. In the case of two-term Wronskian $W(H_n,H_{n+k})$ we give an explicit formula for the asymptotic shape of the zero set as $n\to \infty$. Some empirical asymptotic formulas are given for the zero sets of three and four-term Wronskians.

Extended version with an asymptotic analysis of the 2-term case and some empirical formulas in the 3-term and 4-term cases

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