Monodromy-free Schrödinger operators with quadratically increasing potentials

We derive identities between determinants whose entries are Hermite polynomials. These identities have a combinatorial interpretation in terms of Maya diagrams, partitions and Durfee rectangles, and serve to characterize an equivalence class of rational Darboux transformations. Since the determinants have different orders, we analyze the problem of finding the minimal order determinant in each equivalence class, and describe the solution using an elegant graphical interpretation. The results are applied to provide a more efficient representation for exceptional Hermite polynomials and for rational solutions of the Painlevé IV equation. The latter are expressed in terms of the Okamoto and generalized Hermite polynomials.

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“…As a special case of Theorem , we obtain the following identity; see and the references therein. Corollary Let λ be a partition of length ℓ,

λ^{'}

the conjugate partitions of length

ℓ^{'} = λ_{1}

.…”

Section: Hermite Pseudo‐wronskiansmentioning

confidence: 91%

Durfee Rectangles and Pseudo‐Wronskian Equivalences for Hermite Polynomials

2018

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“…These Wronskian Hermite polynomials appear in the theory of exceptional orthogonal polynomials and the related topic of rational extensions of the quantum harmonic oscillator . These polynomials are well studied.…”

Section: Introductionmentioning

confidence: 99%

Coefficients of Wronskian Hermite polynomials

2020

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We study Wronskians of Hermite polynomials labeled by partitions and use the combinatorial concepts of cores and quotients to derive explicit expressions for their coefficients. These coefficients can be expressed in terms of the characters of irreducible representations of the symmetric group, and also in terms of hook lengths. Further, we derive the asymptotic behavior of the Wronskian Hermite polynomials when the length of the core tends to infinity, while fixing the quotient. Via this combinatorial setting, we obtain in a natural way the generalization of the correspondence between Hermite and Laguerre polynomials to Wronskian Hermite polynomials and Wronskians involving Laguerre polynomials. Lastly, we generalize most of our results to polynomials that have zeros on the p‐star.

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“…If

b (z)

has no real zeros, then L is a Sturm‐Liouville operator on

double-struckR

with quasi‐polynomial eigenfunctions. Exact solvability by polynomials is a very stringent property, which is equivalent to trivial monodromy . In fact, the next proposition proved in Ref.…”

Section: Cyclic Maya Diagrams and Rational Extensions Of The Harmonicmentioning

confidence: 85%

Cyclic Maya diagrams and rational solutions of higher order Painlevé systems

Clarkson

Gómez‐Ullate

2020

Stud Appl Math

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This paper focuses on the construction of rational solutions for the A2n‐Painlevé system, also called the Noumi‐Yamada system, which are considered the higher order generalizations of PIV. In this even case, we introduce a method to construct the rational solutions based on cyclic dressing chains of Schrödinger operators with potentials in the class of rational extensions of the harmonic oscillator. Each potential in the chain can be indexed by a single Maya diagram and expressed in terms of a Wronskian determinant whose entries are Hermite polynomials. We introduce the notion of cyclic Maya diagrams and characterize them for any possible period, using the concepts of genus and interlacing. The resulting classes of solutions can be expressed in terms of special polynomials that generalize the families of generalized Hermite, generalized Okamoto, and Umemura polynomials, showing that they are particular cases of a larger family.

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Monodromy-free Schrödinger operators with quadratically increasing potentials

Cited by 53 publications

References 5 publications

Durfee Rectangles and Pseudo‐Wronskian Equivalences for Hermite Polynomials

Durfee Rectangles and Pseudo‐Wronskian Equivalences for Hermite Polynomials

Coefficients of Wronskian Hermite polynomials

Cyclic Maya diagrams and rational solutions of higher order Painlevé systems

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