Holomorphic Hermite polynomials in two variables

Abstract

Generalizations of the Hermite polynomials to many variables and/or to the complex domain have been located in mathematical and physical literature for some decades. Polynomials traditionally called complex Hermite ones are mostly understood as polynomials in z and z which in fact makes them polynomials in two real variables with complex coefficients. The present paper proposes to investigate for the first time holomorphic Hermite polynomials in two variables. Their algebraic and analytic properties are developed here. While the algebraic properties do not differ too much for those considered so far, their analytic features are based on a kind of non-rotational orthogonality invented by van Eijndhoven and Meyers. Inspired by their invention we merely follow the idea of Bargmann's seminal paper (1961) giving explicit construction of reproducing kernel Hilbert spaces based on those polynomials. "Homotopic" behavior of our new formation culminates in comparing it to the very classical Bargmann space of two variables on one edge and the aforementioned Hermite polynomials in z and z on the other. Unlike in the case of Bargmann's basis our Hermite polynomials are not product ones but factorize to it when bonded together with the first case of limit properties leading both to the Bargmann basis and suitable form of the reproducing kernel. Also in the second limit we recover standard results obeyed by Hermite polynomials in z and z.1991 Mathematics Subject Classification. Primary 33A65, 44E22, 47B32 ; Secondary 33C45 . Key words and phrases. Hermite polynomials in two complex variables, reproducing kernel Hilbert space, van Eijndhoven and Meyers type orthogonality, creation and annihilation operators.

Coherence, Squeezing and Entanglement: An Example of Peaceful Coexistence

“…The holomorphic Hermite polynomials in one and two variables, as well as holomorphic Hermite functions determined by them, will be our main tool extensively used in the next Sections to construct coherent states. This Section serves as a kind of technical introduction and revokes the formulae derived and proved in [4] and [20].…”

“…On the other hand the limit α → 0+ leads to results which forbide to construct any kind of CSs being well-defined within our scheme. This is because performing this limit breaks down the fundamental condition (2) and, consequently, the normalizability of CSs [20,49]. Nevertheless, the polynomials H m,n (z,z), which arise in the limit α → 0+ of the two variable generalization of the van Eijndhoven-Meiers picture (see the Section 2.2), have found plenty of interesting applications: to mention investigation of their relation with the entangled (in particular EPR) states begun more than 20 years ago [12], continued in [13], and still being the subject of extensive research (cf.…”

“…Using (16) the Hermite polynomials in two variables can be expressed, like it is shown in [20,Eq. (13)], in terms of the Hermite polynomials in a single variable (9)…”

“…Here we want to emphasize that (15) defines polynomials in two complex variables with real coefficients while H m,n (z, z) are polynomials in two real variables x = Re z and y = Im z with complex coefficients. This may be somehow confusing when the term "complex Hermite polynomials" appears for the latter, for more discussion see the introductory section in [20]. valid for 0 < α < 1, [20,Eq.…”

“…This may be somehow confusing when the term "complex Hermite polynomials" appears for the latter, for more discussion see the introductory section in [20]. valid for 0 < α < 1, [20,Eq. (19)].…”