Complex Analysis and Operator Theory

, Volume 13, Issue 2, pp 351–374 | Cite as

Holomorphic Hermite Functions in Segal–Bargmann Spaces

  • Hiroyuki ChiharaEmail author


We study systems of holomorphic Hermite functions in the Segal–Bargmann spaces, which are Hilbert spaces of entire functions on the complex Euclidean space, and are determined by the Bargmann-type integral transform on the real Euclidean space. We prove that for any positive parameter which is strictly smaller than the minimum eigenvalue of the positive Hermitian matrix associated with the transform, one can find a generator of holomorphic Hermite functions whose annihilation and creation operators satisfy canonical commutation relations. In other words, we find the necessary and sufficient conditions so that some kinds of entire functions can be such generators. Moreover, we also study the complete orthogonality, the eigenvalue problems and the Rodrigues formulas.


Bargmann transform Segal–Bargmann spaces Holomorphic Hermite functions 

Mathematics Subject Classification

Primary 33C45 Secondary 46E20 46E22 35S30 


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of EducationUniversity of the RyukyusNishiharaJapan

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