A pair of commuting hypergeometric operators on the complex plane and bispectrality

Vladimir F. Molchanov,Yury A. Neretin

doi:10.4171/jst/349

Vladimir F. Molchanov, Yury A. Neretin

Open Access

https://doi.org/10.4171/jst/349

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Abstract

We consider the standard hypergeometric differential operator $D$ regarded as an operator on the complex plane $C$ and the complex conjugate operator $\overline D$. These operators formally commute and are formally adjoint one to another with respect to an appropriate weight. We find conditions when they commute in the Nelson sense and write explicitly their joint spectral decomposition. It is determined by a two-dimensional counterpart of the Jacobi transform (synonyms: generalized Mehler--Fock transform, Olevskii transform). We also show that the inverse transform is an operator of spectral decomposition for a pair of commuting difference operators defined in terms of shifts in imaginary direction.

Highlights

We notice that functions, which can be regarded as higher hypergeometric functions 4F3C of the complex field, arise in a natural way in the work of Ismagilov [20] as analogs of the Racah coefficients for unitary representations of the Lorentz group SL(2, C)
We get a family of functions depending of 4 parameters τij, for generic a, b, c this formula gives all multivalued solutions near z = 0
We have the same asymptotics (4.4)–(4.5), we only must write the coefficients of the form A(0, τ ), B(0, τ ), C(0, τ ), D(0, τ ) in (4.4)–(4.5)

Summary

Introduction

To prove the necessary conditions of self-adjointness in Theorem 1.1 we analyze common generalized eigenfunctions of D, D for (a, b) ∈/ Π and after a natural selection we reduce a set of possible candidates to a finite family We notice that functions, which can be regarded as higher hypergeometric functions 4F3C of the complex field, arise in a natural way in the work of Ismagilov [20] as analogs of the Racah coefficients for unitary representations of the Lorentz group SL(2, C) (see, a continuation in [5]) It seems that our problem can be a representative of some family of spectral problems, but it is too early to claim something certainly.

The hypergeometric function of the complex field

Nonexistence of commuting self-adjoint extensions

Symmetry of differential operators

Asymptotics of the kernel in the parameters

Symmetry of difference operators

10. Domains of self-adjointness