D-Pseudo-Bosons, Complex Hermite Polynomials, and Integral Quantization

Ali, S. Twareque; Багарелло, Фабио; Gazeau, Jean‐Pierre

doi:10.3842/sigma.2015.078

Cited by 11 publications

(14 citation statements)

References 47 publications

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“…Typical providers are indeterminate moment problems or rather orthonormal polynomials coming from them. If (Φ n ) n is such a sequence of polynomials then the well known consequence is that it satisfies (2). As already shown any of the orthogonality measures appearing in this problem works well for the resolution of the identity to be satisfied.…”

Section: 5mentioning

confidence: 89%

“…Let us recall that, according to our definition of coherent states evolved in Section 1.4 and starting from the formula (3), the basic requirement for some states to be called coherent is to be provided with (Φ n ) n satisfying (2). Now the Zaremba construction guarantees existence of the Segal-Bargmann transform, the property which is historically and not too rigorously identified with the overcompleteness and/or the resolution of the identity.…”

Section: Hsz Css -Holomorphic Hermite Polynomials Perspectivementioning

confidence: 99%

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Coherence, Squeezing and Entanglement: An Example of Peaceful Coexistence

Górska

Horzela

Szafraniec

2018

Springer Proceedings in Physics

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After exhaustive inspection of bosonic coherent states appearing in physical literature two of us, Horzela and Szafraniec, came in 2012 to the reasonably general definition which relies exclusively on reproducing kernels. The basic feature of coherent states, which is the resolution of the identity, is preserved though it now achieves advantageous form of the Segal-Bargmann transform. It turns out that the aforesaid definition is not only extremely economical but also puts under a common umbrella typical coherent states as well as those which are squeezed and entangled. We examine the case here on the groundwork of holomorphic Hermite polynomials in one and two variables. An interesting side of this story is how some limit procedure allows disentangling.

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Section: 5mentioning

confidence: 89%

Section: Hsz Css -Holomorphic Hermite Polynomials Perspectivementioning

confidence: 99%

Coherence, Squeezing and Entanglement: An Example of Peaceful Coexistence

Górska

Horzela

Szafraniec

2018

Springer Proceedings in Physics

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“…Laguerre 2D polynomials L m,n (z, z ) with two, in general, independent complex variables z and z were introduced in [7][8][9][10][11][12] by (similar or more general objects with other names and notations were defined in [13][14][15][16][17][18][19][20][21][22][23][24]) The differentiation of the Laguerre 2D polynomials provides again Laguerre 2D polynomials ∂ ∂z L m,n (z, z ) = m L m−1,n (z, z ), ∂ ∂z L m,n (z, z ) = n L m,n−1 (z, z ), (1.5) and, furthermore, the Laguerre 2D polynomials satisfy the following recurrence relations L m+1,n (z, z ) = z L m,n (z, z ) − n L m,n−1 (z, z ), L m,n+1 (z, z ) = z L m,n (z, z ) − m L m−1,n (z, z ), (1.6) as was derived in [7][8][9] and as can be easily seen from (1.1) or (1.2). The Laguerre 2D polynomials (1.2) are related to the generalized Laguerre (or Laguerre-Sonin) 1 polynomials L ν n (u) by that explains the given name.…”

Section: Introductionmentioning

confidence: 99%

Generating Functions for Products of Special Laguerre 2D and Hermite 2D Polynomials

Wünsche¹

2015

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The bilinear generating function for products of two Laguerre 2D polynomials L m,n (z, z ) with different arguments is calculated. It corresponds to the formula of Mehler for the generating function of products of two Hermite polynomials. Furthermore, the generating function for mixed products of Laguerre 2D and Hermite 2D polynomials and for products of two Hermite 2D polynomials is calculated. A set of infinite sums over products of two Laguerre 2D polynomials as intermediate step to the generating function for products of Laguerre 2D polynomials is evaluated but these sums possess also proper importance for calculations with Laguerre polynomials. With the technique of SU (1, 1) operator disentanglement some operator identities are derived in an appendix. They allow to calculate convolutions of Gaussian functions combined with polynomials in one-and two-dimensional case and are applied to evaluate the discussed generating functions.

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“…This is we abandon the term "complex polynomials". 3 Differentiation as applied to polynomials can also be thought of as an algebraic property.…”

Section: Algebraic Propertiesmentioning

confidence: 99%

Holomorphic Hermite polynomials in two variables

Górska

Horzela

Szafraniec

2019

Journal of Mathematical Analysis and Applications

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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.

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D-Pseudo-Bosons, Complex Hermite Polynomials, and Integral Quantization

Cited by 11 publications

References 47 publications

Coherence, Squeezing and Entanglement: An Example of Peaceful Coexistence

Coherence, Squeezing and Entanglement: An Example of Peaceful Coexistence

Generating Functions for Products of Special Laguerre 2D and Hermite 2D Polynomials

Holomorphic Hermite polynomials in two variables

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