Explicit formulas for reproducing kernels of generalized Bargmann spaces on Cn

Abstract

We introduce a class of generalized Bargmann spaces on Cn for which we establish explicit formulas of their reproducing kernels. Some applications of the obtained formulas are given.

Fock spaces, Landau operators and the time-harmonic Maxwell equations

“…Also, the generalization of the theory of Segal-Bargmann spaces during the last decade by Askour et al (cf [3]) and Vasilevski (cf [35]) led recently to ubiquitous developments in the theory of Gabor-Window Fourier analysis [1,2,5]. From the border view of the Heisenberg group, it was recently shown in [13] that the Schrödinger representation and alike play an important role in the study of pseudo-differential operators establishing structural properties between the Weyl calculus and the Landau-Weyl calculus.…”

“…shall be interpreted as the monogenic counterparts for the real Bargmann spaces (also called Segal-Bargmann, Fock or Fischer spaces, see [4,10,28,30]) and poly-Bargmann spaces (also called poly-Fock or generalized Bargmann spaces, [3,35]), respectively. These sorts of spaces are proper subspaces of the so-called poly-monogenic functions with respect to the C ∞ -topology (cf [26]).…”

On Concrete Spectral Properties of a Twisted Laplacian Associated with a Central Extension of the Real Heisenberg Group

“…The proof for ] = 0 is contained in [1,4,5]. For arbitrary ], the proof can be handled in a similar way or making use of the key observation that in 2 (C , ), the operators Δ ], and Δ 0, are unitary equivalents and we have…”

“…The key observation is contained in the identity (65) which will serve as an outline of the proof of Proposition 15 as well as the proofs of the assertions below, taking into account the well-established results for Δ 0, (see [1,[4][5][6][7] and the references therein).…”

“…It has been considered and studied from different point of views in physics as in mathematics [1][2][3][4][5][6][7]. It goes back to L. D. Landau (for = 1) and plays an important role in many different contexts such as Feynman path integral (in Feynman-Kac formula), oscillatory stochastic integral, and theory of lattices electrons in uniform magnetic field.…”