| Contents |
| Exceptional Hermite polynomials are complete systems of orthogonal polynomials that satisfy a Sturm-Liouville problem in the real line. They differ from Hermite polynomials in that there are a finite number of gaps in their degree sequence. We will show a complete classification of such polynomial systems by using Darboux-Crum transformations and showing that the associated potentials are monodromy free.
A convenient representation of exceptional Hermite polynomials is through Wronskian determinants of certain sequences of classical Hermite polynomials, and we will discuss some properties of their real and complex zeros. Particular cases of exceptional Hermite polynomials appear in the construction of rational solutions to Nonlinear Schr\"odinger's Equation, and we will try to illustrate this connection. |
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