|
SIGMA 11 (2015), 078, 23 pages arXiv:1509.03822
https://doi.org/10.3842/SIGMA.2015.078
${\mathcal D}$-Pseudo-Bosons, Complex Hermite Polynomials, and Integral Quantization
S. Twareque Ali a, Fabio Bagarello b and Jean Pierre Gazeau cd
a) Department of Mathematics and Statistics, Concordia University, Montréal, Québec, Canada H3G 1M8
b) Dipartimento di Energia, ingegneria dell'Informazione e modelli Matematici, Scuola Politecnica, Università di Palermo, I-90128 Palermo, and INFN, Torino, Italy
c) APC, UMR 7164, Univ Paris Diderot, Sorbonne Paris-Cité, 75205 Paris, France
d) Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro, 22290-180 Rio de Janeiro, Brazil
Received March 28, 2015, in final form September 21, 2015; Published online October 01, 2015
Abstract
The ${\mathcal D}$-pseudo-boson formalism is illustrated with two examples. The first one involves deformed complex Hermite polynomials built using finite-dimensional irreducible representations of the group ${\rm GL}(2,{\mathbb C})$ of invertible $2 \times 2$ matrices with complex entries. It reveals interesting aspects of these representations. The second example is based on a pseudo-bosonic generalization of operator-valued functions of a complex variable which resolves the identity. We show that such a generalization allows one to obtain a quantum pseudo-bosonic version of the complex plane viewed as the canonical phase space and to understand functions of the pseudo-bosonic operators as the quantized versions of functions of a complex variable.
Key words:
pseudo-bosons; coherent states; quantization; complex Hermite polynomials; finite group representation.
pdf (581 kb)
tex (69 kb)
References
-
Abreu L.D., Sampling and interpolation in Bargmann-Fock spaces of polyanalytic functions, Appl. Comput. Harmon. Anal. 29 (2010), 287-302, arXiv:0901.4386.
-
Abreu L.D., Feichtinger H.G., Function spaces of polyanalytic functions, in Harmonic and Complex Analysis and its Applications, Editor A. Vasil'ev, Trends Math., Birkhäuser/Springer, Cham, 2014, 1-38.
-
Abreu L.D., Gröchenig K., Banach Gabor frames with Hermite functions: polyanalytic spaces from the Heisenberg group, Appl. Anal. 91 (2012), 1981-1997, arXiv:1012.4283.
-
Agorram F., Benkhadra A., El Hamyani A., Ghanmi A., Complex Hermite functions as Fourier-Wigner transform, arXiv:1506.07084.
-
Ali S.T., Antoine J.P., Gazeau J.P., Coherent states, wavelets, and their generalizations, 2nd ed., Theoretical and Mathematical Physics, Springer, New York, 2014.
-
Ali S.T., Bagarello F., Gazeau J.P., Quantizations from reproducing kernel spaces, Ann. Physics 332 (2013), 127-142, arXiv:1212.3664.
-
Ali S.T., Bhattacharyya T., Roy S.S., Coherent states on Hilbert modules, J. Phys. A: Math. Theor. 44 (2011), 275202, 16 pages, arXiv:1007.0798.
-
Ali S.T., Ismail M.E.H., Shah N.M., Deformed complex Hermite polynomials, arXiv:1410.3908.
-
Bagarello F., Examples of pseudo-bosons in quantum mechanics, Phys. Lett. A 374 (2010), 3823-3827, arXiv:1007.4349.
-
Bagarello F., Pseudobosons, Riesz bases, and coherent states, J. Math. Phys. 51 (2010), 023531, 10 pages, arXiv:1001.1136.
-
Bagarello F., From self to non self-adjoint harmonic oscillators: physical consequences and mathematical pitfalls, Phys. Rev. A 88 (2013), 032120, 5 pages, arXiv:1309.5065.
-
Bagarello F., More mathematics for pseudo-bosons, J. Math. Phys. 54 (2013), 063512, 11 pages, arXiv:1309.0677.
-
Bagarello F., Deformed canonical (anti-)commutation relations and non self-adjoint Hamiltonians, in Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects, Editors F. Bagarello, J.P. Gazeau, F.H. Szafraniec, M. Znojil, John Wiley & Sons, Inc, Hoboken, NJ, 2015, 121-188.
-
Bagarello F., Fring A., Non-self-adjoint model of a two-dimensional noncommutative space with an unbound metric, Phys. Rev. A 88 (2013), 042119, 6 pages, arXiv:1310.4775.
-
Baldiotti M., Fresneda R., Gazeau J.P., Three examples of covariant integral quantization, PoS Proc. Sci. (2014), PoS(ICMP2013), 003, 18 pages.
-
Balogh F., Shah N.M., Ali S.T., On some families of complex Hermite polynomials and their applications to physics, in Operator Algebras and Mathematical Physics, Operator Theory: Advances and Applications, Vol. 247, Editors T. Bhattacharyya, M.A. Dritschel, Birkhäuser, Basel, 2015, 157-171, arXiv:1309.4163.
-
Bergeron H., Gazeau J.P., Integral quantizations with two basic examples, Ann. Physics 344 (2014), 43-68, arXiv:1308.2348.
-
Cahill K.E., Glauber R.J., Ordered expansion in boson amplitude operators, Phys. Rev. 177 (1969), 1857-1881.
-
Cotfas N., Gazeau J.P., Górska K., Complex and real Hermite polynomials and related quantizations, J. Phys. A: Math. Theor. 43 (2010), 305304, 14 pages, arXiv:1001.3248.
-
Cuntz J., Simple $C^*$-algebras generated by isometries, Comm. Math. Phys. 57 (1977), 173-185.
-
Davies E.B., Pseudo-spectra, the harmonic oscillator and complex resonances, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999), 585-599.
-
Davies E.B., Linear operators and their spectra, Cambridge Studies in Advanced Mathematics, Vol. 106, Cambridge University Press, Cambridge, 2007.
-
Davies E.B., Kuijlaars A.B.J., Spectral asymptotics of the non-self-adjoint harmonic oscillator, J. London Math. Soc. 70 (2004), 420-426.
-
Gazeau J.P., Heller B., Positive-operator valued measure (POVM) quantization, Axioms 4 (2015), 1-29, arXiv:1408.6090.
-
Ghanmi A., A class of generalized complex Hermite polynomials, J. Math. Anal. Appl. 340 (2008), 1395-1406, arXiv:0704.3576.
-
Gradshteyn I.S., Ryzhik I.M., Table of integrals, series, and products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007.
-
Gröchenig K., Lyubarskii Y., Gabor (super)frames with Hermite functions, Math. Ann. 345 (2009), 267-286, arXiv:0804.4613.
-
Haimi A., Hedenmalm H., The polyanalytic Ginibre ensembles, J. Stat. Phys. 153 (2013), 10-47, arXiv:1106.2975.
-
Ismail M.E.H., Analytic properties of complex Hermite polynomials, Trans. Amer. Math. Soc., to appear.
-
Ismail M.E.H., Simeonov P., Complex Hermite polynomials: their combinatorics and integral operators, Proc. Amer. Math. Soc. 143 (2015), 1397-1410.
-
Ismail M.E.H., Zeng J., A combinatorial approach to the 2D-Hermite and 2D-Laguerre polynomials, Adv. in Appl. Math. 64 (2015), 70-88.
-
Ismail M.E.H., Zeng J., Two variable extensions of the Laguerre and disc polynomials, J. Math. Anal. Appl. 424 (2015), 289-303.
-
Ismail M.E.H., Zhang R., Classes of bivariate orthogonal polynomials, arXiv:1502.07256.
-
Itô K., Complex multiple Wiener integral, Japan J. Math. 22 (1952), 63-86.
-
Magnus W., Oberhettinger F., Soni R.P., Formulas and theorems for the special functions of mathematical physics, Die Grundlehren der mathematischen Wissenschaften, Vol. 52, 3rd ed., Springer-Verlag, New York, 1966.
-
Pedersen G.K., Analysis now, Graduate Texts in Mathematics, Vol. 118, Springer-Verlag, New York, 1989.
-
Szegő G., Orthogonal polynomials, American Mathematical Society, Colloquium Publications, Vol. 23, 4th ed., Amer. Math. Soc., Providence, R.I., 1975.
-
Trifonov D.A., Pseudo-boson coherent and Fock states, in Trends in Differential Geometry, Complex Analysis and Mathematical Physics, World Sci. Publ., Hackensack, NJ, 2009, 241-250, arXiv:0902.3744.
-
Vasilevski N.L., Poly-Fock spaces, in Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, Birkhäuser, Basel, 2000, 371-386.
-
Wünsche A., Laguerre $2$D-functions and their application in quantum optics, J. Phys. A: Math. Gen. 31 (1998), 8267-8287.
-
Young R.M., On complete biorthogonal systems, Proc. Amer. Math. Soc. 83 (1981), 537-540.
|
|