Non-trivial 1d and 2d Segal–Bargmann transforms

ABSTRACT

Special generating functions and Mehler's formula for the univariate complex Hermite polynomials are obtained and next employed to introduce and study some one- and two-dimensional integral transforms of Segal–Bargmann type in the framework of some specific functional Hilbert spaces; including the so-called generalized Bargmann–Fock spaces that are realized as $L^2$-eigenspaces of a special magnetic Schrödinger operator.

KEYWORDS:

• Univariate complex Hermite polynomials
• generating function
• Mehler's fomulas
• magnetic Schrödinger operator
• generalized Bargmann–Fock spaces
• integral transform of Segal–Bargmann type
• $2d$-fractional Fourier transform

AMS CLASSIFICATIONS:

• 35A22
• 44A15
• 33C45
• 42C05
• 42A38
• 32A10

Acknowledgements

The authors would like to thank the anonymous referees for helpful comments. They acknowledge their indebtedness to Prof. Ahmed Sebbar for his encouragements. The activate members of ‘Ahmed Intissar seminar's’ in Rabat are gratefully acknowledged.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Allal Ghanmi  http://orcid.org/0000-0003-0764-5576