New generalized Hermite polynomials with three variables obtained via quantum optics method and their applications

Generalized Hermite polynomials

The paper gives a review of recent progress in the classification of monodromy-free Schr\"odinger operators with rational potentials. We concentrate on a class of potentials constituted by generalized Hermite polynomials. These polynomials defined as Wronskians of classic Hermite polynomials appear in a number of mathematical physics problems as well as in the theory of random matrices and 1D SUSY quantum mechanics. Being quadratic at infinity, those potentials demonstrate localized oscillatory behavior near the origin. We derive an explicit condition of non-singularity of the corresponding potentials and estimate a localization range with respect to indices of polynomials and distribution of their zeros in the complex plane. It turns out that 1D SUSY quantum non-singular potentials come as a dressing of the harmonic oscillator by polynomial Heisenberg algebra ladder operators. To this end, all generalized Hermite polynomials are produced by appropriate periodic closure of this algebra which leads to rational solutions of the Painlev\'e IV equation. We discuss the structure of the discrete spectrum of Schr\"odinger operators and its link to the monodromy-free condition.

Although the origin of quantum optics can be traced back to the beginning of the 20th century, when the fundamental ideas about the quantum nature of the interaction between light and matter were put forward, the splendid blossoming of this part of physics began half a century later, after the invention of masers and lasers. It is remarkable that after another half a century the tree of quantum optics is not only very strong and spreading, but all its branches continue to grow, showing new beautiful blossoms and giving very useful fruits.

In the first half of the article, we present two theorems which give, as special cases, a number of new classes of generating functions for the Laguerre polynomial. These formulae extend the recent results of Carlitz [2] and others. The latter part of our work deals with two theorems involving new generating functions for the Hermite and generalized Hermite polynomials, thus generalizing some well-known expansions. The method of proof adopted in this paper differs from that of previous workers.

The associated Laguerre polynomials LH"(x) and the Hermite polynomials #*(£) occupy a distinctive place in quantum mechanics.A wellestablished relationship exists between Hk(£) and Ln±'b(x); this can be further exploited to obtain generalized Hermite polynomials Hk*(%).The latter, relevant to spatial harmonic oscillators in curvilinear coordinates, are essentially the polynomials obtained by Stroud and Secrest [11] based on transformations of numerical quadratures.The tables presented here are zeros and Gaussian weights of {Ln"(x)\ with a = -0.5(1.0)3.5, re = 4, 8, 16, 32; and those of {#/(!)! with A = 0(1.0)4.0,k = 8, 16, 32, 64.In both cases the results are believed to be correct to one unit in the 25th decimal place.The notations for Lna(x) and Hk(%) employed here are those in Szegö [12] and adopted in Erdélyi et al. [5]; namely a» ^w-("íj.«-»;«+^)-¿(:í:)(x)m/ml, lk/2] (1.2) #*({) = fc!E (-l)m(2£)*-27m!(fc -2m)! \Lna{x)} thus defined forms an orthogonal set in [0, » ) with respect to the weight function e~xx"; Hk(£) are similarly orthogonal in ( -», + oo ) with respect to e~£ .Many physicists, notably Condon and Shortley [3], employ (1.3) Lna(x) = re! ( j iPj( -re + a; a + 1; x) = a polynomial of degree n -a, while many others, e.g., Morse and Feshbach [8] have (1.4)Ln"(x) = (" + «)! (n + a) iFi{-n; a + 1; x).The mathematical statistics literature often employs the notation [km (1.5) Hk(S) = 2~*/2fc!£ (-l)m(V2H)k-2m/ml (fc -2m)! which would correspond to Hek(i) in Erdélyi et al., and /&*(£) in Davis [4], and the corresponding weighting function is e_i /2.

Hermite polynomials with many variables and many indices play a crucial role within the framework of phase-space formulation of classical or quantum mechanics. A generalized operational formalism useful to handle these polynomials is discussed and also a new set of creation–annihilation operators associated to the phase-space harmonic oscillator orthogonal functions is introduced. The Fourier transform of these orthogonal functions is studied by discussing the analogies with the ordinary case. Finally, the integral representation of the generalized Hermite polynomials and a simple technique to deal with the generalized heat equation are discussed.

Generalized Hermite polynomials in superspace as eigenfunctions of the supersymmetric rational CMS model

Theory of generalized hermite polynomials

This paper studies a suitably normalized set of generalized Hermite polynomials and sets down a relevant Mehler formula, Rodrigues formula, and generalized translation operator. Weighted generalized Hermite polynomials are the eigenfunctions of a generalized Fourier transform which satisfies an F. and M. Riesz theorem on the absolute continuity of analytic measures. The Bose-like oscillator calculus, which generalizes the calculus associated with the quantum mechanical simple harmonic oscillator, is studied in terms of these polynomials.

Model of the extended Dunkl oscillator based on Milovanović generalized Hermite polynomials on radial rays is discussed. Simple explicit realization of creation and annihilation operators in terms of difference-differential operators, coherent states are investigated.

Time evolution of coupled spin systems in a generalized Wigner representation

Employing the general ordering theorem (GOT), operational methods and incomplete 2-D Hermite polynomials, we derive the t-ordered expansion of Fock space projectors. Using the result, the general ordered form of the coherent state projectors is obtained. This indeed gives a new integration formula regarding incomplete 2-D Hermite polynomials. In addition, the orthogonality relation of the incomplete 2-D Hermite polynomials is derived to resolve Dattoli's failure.

The Painlevé-IV equation has three families of rational solutions generated by the generalized Hermite polynomials. Each family is indexed by two positive integers $m$ and $n$. These functions have applications to nonlinear wave equations, random matrices, fluid dynamics, and quantum mechanics. Numerical studies suggest the zeros and poles form a deformed $n\times m$ rectangular grid. Properly scaled, the zeros and poles appear to densely fill certain curvilinear rectangles as $m,n\to \infty $ with $r:=m/n$ a fixed positive real number. Generalizing a method of Bertola and Bothner [2] used to study rational Painlevé-II functions, we express the generalized Hermite rational Painlevé-IV functions in terms of certain non-Hermitian orthogonal polynomials. Using the Deift–Zhou nonlinear steepest-descent method, we asymptotically analyze the associated Riemann–Hilbert problem in the limit $n\to \infty $ with $m=r\cdot n$ for $r$ fixed. We obtain an explicit characterization of the boundary curve and determine the leading-order asymptotic expansion of the rational Painlevé-IV functions associated to generalized Hermite polynomials in the pole-free region.

In this paper, we study Hahn’s problem with respect to a Dunkl-perturbed raising operator. More precisely, we prove that, up to a dilation, the generalized Hermite polynomials are the only Tμ,α-classical symmetric orthogonal polynomials, where Tμ,α=Tμ+αtI, α∈C∖{0} and I denotes the identity operator on the space of polynomials with complex coefficients. The argument uses an operator product rule for Tμ, duality for the associated functionals, and a symmetry-enforced identification together with matching three-term recurrences. The result provides an operator-theoretic Hahn-type characterization that complements semiclassical Pearson-equation descriptions and clarifies the effect of the raising perturbation αtI.

Summation formulae for Gould–Hopper generalized Hermite polynomials