Mehler Formula Research Articles

Closed-Form Propagator of the Calogero Model.

We present a generalization of the Mehler kernel, providing the exact analytic time evolution of the Calogero model that describes bosons or fermions propagating on the real line with inverse-square two-body interactions and harmonic confinement. The key to our result is a simple relation between the exact propagator of the Calogero model and a Baker-Akhiezer function, together with explicit combinatorial formulas for the latter function.

Applications of the Finite Operator〖 〗_3 E_2 (■(q^(-N),a,b@c,d);q,-fθ) for the Polynomials B_n (a,b,c,d,f,x,y|q)

In this work, polynomials and the finite q-exponential operator are constructed. The operator is used to combine an operator proof of the generating function with its extension, Mehler's formula with its extension and Roger's formula for the polynomials . The generating function with its extension, Mehler's formula with its extension and Rogers formula for Al-Salam-Carlitz polynomials are deduced by giving special values to polynomials .

De Bruijn identities in different Markovian channels

De Bruijn's identity in information theory states that if u is the solution of the heat equation, then the time derivative of the Shannon entropy for this solution is equal to the amount of Fisher information at u. In this article, we show how this identity changes if we replace the heat channel by the Fokker Planck, or passing from Fokker Planck to Ornstein-Uhlenbeck channels. Through these passages we investigate the different properties of these solutions. We exclusively dissect different properties of Ornstein-Uhlenbeck semigroup given by the Mehler formula expression. For more information see https://ejde.math.txstate.edu/Volumes/2023/12/abstr.html

Two generalizations of Mehler's formula in white noise analysis

ABSTRACT Mehler's formula is an important tool in Gaussian analysis. In this article, we study two generalizations of Mehler's formula for the Ornstein–Uhlenbeck semigroup, i.e. the semigroup generated by the number operator. The first generalization leads to transformation groups which have as infinitesimal generator a perturbation of the number operator with suitable integral kernel operators, which are well studied in white noise analysis. For the second one, we characterize the complex Hida measures for which a version of Mehler's formula for the Ornstein–Uhlenbeck semigroup can be extended to. We apply this result to the Feynman integrand for a quadratic potential. Here the time independent eigenstates of the considered transformation groups and the time evolution of eigenvalues are provided.

Mehler kernel approach to Fourier ultra-hyperfunctions

ABSTRACT In this paper we shall characterize the space of Fourier ultra-hyperfunctions by the Mehler kernel method. That is, we show that any Fourier ultra-hyperfunctions are characterized as initial values of the solutions of Hermite heat equation.

Multivariate holomorphic Hermite polynomials

We introduce holomorphic Hermite polynomials in n complex variables that generalize the Hermite polynomials in n real variables introduced by Hermite in the late 19th century. We discuss cases in which these polynomials are orthogonal and construct a reproducing kernel Hilbert space related to one such orthogonal family. We also introduce a multivariate analog of the Ito polynomials. We show how these multivariate polynomials generalize the univariate complex Hermite and Ito polynomials. Generating functions, orthogonality relations, Rodrigues formulas, recurrence and linearization relations, and operator formulas are also derived for these multivariate holomorphic Hermite and Ito polynomials. A Kibble–Slepian formula and a Mehler-type formula for the multivariate Ito polynomials are established.

Bloch-Messiah reduction for twin beams of light

We study the Bloch-Messiah reduction of parametric downconversion of light in the pulsed regime with a nondegenerate phase matching providing generation of twin beams. We find that in this case every squeezing eigenvalue has multiplicity at least two. We discuss the problem of ambiguity in the definition of the squeezing eigenmodes in this case and develop two approaches to unique determination of the latter. First, we show that the modal functions of the squeezing eigenmodes can be tailored from the Schmidt modes of the signal and idler beams. Alternatively, they can be found as a solution of an eigenvalue problem for an associated Hermitian squeezing matrix. We illustrate the developed theory by an example of frequency non-degenerate collinear twin beams generated in beta barium borate crystal. On this example we demonstrate how the squeezing eigenmodes can be approximated analytically on the basis of the Mehler's formula, extended to complex kernels. We show how the multiplicity of the eigenvalues and the structure of the eigenmodes are changed when the phase matching approaches the degeneracy in frequency.

Non-trivial 1d and 2d Segal–Bargmann transforms

ABSTRACTSpecial generating functions and Mehler's formula for the univariate complex Hermite polynomials are obtained and next employed to introduce and study some one- and two-dimensional integral transforms of Segal–Bargmann type in the framework of some specific functional Hilbert spaces; including the so-called generalized Bargmann–Fock spaces that are realized as -eigenspaces of a special magnetic Schrödinger operator.

Some homogeneous q-difference operators and the associated generalized Hahn polynomials

In this paper, we first construct the homogeneous $q$-shift operator $\widetilde{E}(a,b;D_{q})$ and the homogeneous $q$-difference operator $\widetilde{L}(a,b; \theta_{xy})$. We then apply these operators in order to represent and investigate generalized Cauchy and a general form of Hahn polynomials. We derive some $q$-identities such as: generating functions, extended generating functions, Mehler's formula and Roger's formula for these $q$-polynomials.

Generalized Mehler formula for time-dependent non-selfadjoint quadratic operators and propagation of singularities

We study evolution equations associated to time-dependent dissipative non-selfadjoint quadratic operators. We prove that the solution operators to these non-autonomous evolution equations are given by Fourier integral operators whose kernels are Gaussian tempered distributions associated to non-negative complex symplectic linear transformations, and we derive a generalized Mehler formula for their Weyl symbols. Some applications to the study of the propagation of Gabor singularities (characterizing the lack of Schwartz regularity) for the solutions to non-autonomous quadratic evolution equations are given.

Mehler's formulas for the univariate complex Hermite polynomials and applications

We give 2 widest Mehler's formulas for the univariate complex Hermite polynomials , by performing double summations involving the products and . They can be seen as the complex analogues of the classical Mehler's formula for the real Hermite polynomials. The proof of the first one is based on a generating function giving rise to the reproducing kernel of the generalized Bargmann space of level m. The second Mehler's formula generalizes the one appearing as a particular case of the so‐called Kibble‐Slepian formula. The proofs we present here are direct and more simpler. Moreover, direct applications are given and remarkable identities are derived.

On a system of partial differential equations and the bivariate Hermite polynomials

Using the theory of analytic functions of several complex variables, we prove that if an analytic function in several variables satisfies a system of partial differential equations, then, it can be expanded in terms of the product of the bivariate Hermite polynomials. This expansion theorem allows us to develop a systematic method to prove the identities involving the bivariate Hermite polynomials. With this expansion theorem, we can easily derive, among others, the Mehler formula, Nielsen's formulas, Doetsch's formula, the addition formula, Weisner's formulas, Carlitz's formulas for the bivariate Hermite polynomials.

– OFF-DIAGONAL ESTIMATES FOR THE ORNSTEIN–UHLENBECK SEMIGROUP: SOME POSITIVE AND NEGATIVE RESULTS

We investigate $L^{p}(\unicode[STIX]{x1D6FE})$–$L^{q}(\unicode[STIX]{x1D6FE})$ off-diagonal estimates for the Ornstein–Uhlenbeck semigroup $(e^{tL})_{t>0}$. For sufficiently large $t$ (quantified in terms of $p$ and $q$), these estimates hold in an unrestricted sense, while, for sufficiently small $t$, they fail when restricted to maximal admissible balls and sufficiently small annuli. Our counterexample uses Mehler kernel estimates.

On the integral kernels of derivatives of the Ornstein–Uhlenbeck semigroup

This paper presents a closed-form expression for the integral kernels associated with the derivatives of the Ornstein–Uhlenbeck semigroup [Formula: see text]. Our approach is to expand the Mehler kernel into Hermite polynomials and apply the powers [Formula: see text] of the Ornstein–Uhlenbeck operator to it, where we exploit the fact that the Hermite polynomials are eigenfunctions for [Formula: see text]. As an application we give an alternative proof of the kernel estimates by Ref. 10, making all relevant quantities explicit.

Combinatorial and analytic properties of the n-dimensional Hermite polynomials

We investigate some combinatorial and analytic properties of the n-dimensional Hermite polynomials introduced by Hermite in the late 19-th century. We derive combinatorial interpretations and recurrence relations for these polynomials. We also establish a new linear generating function and a Kibble–Slepian formula for the n-dimensional Hermite polynomials which generalize the Kibble–Slepian formula for the univariate Hermite polynomials and the Poisson kernel (Mehler formula) for the n-dimensional Hermite polynomials.

Slice Fourier transform and convolutions

Recently the construction of various integral transforms for slice monogenic functions has gained a lot of attention. In line with these developments, the article at hand introduces the slice Fourier transform. In the first part, the kernel function of this integral transform is constructed using the Mehler formula. An explicit expression for the integral transform is obtained and allows for the study of its properties. In the second part, two kinds of corresponding convolutions are examined: Mustard convolutions and convolutions based on generalised translation operators. The paper finishes by demonstrating the connection between both.

Quantum theory with an energy operator defined as a quartic form of the momentum

Quantum theory of the non-harmonic oscillator defined by the energy operator proposed by Yurke and Buks (2006) is presented. Although these authors considered a specific problem related to a model of transmission lines in a Kerr medium, our ambition is not to discuss the physical substantiation of their model. Instead, we consider the problem from an abstract, logically deductive, viewpoint. Using the Yurke–Buks energy operator, we focus attention on the imaginary-time propagator. We derive it as a functional of the Mehler kernel and, alternatively, as an exact series involving Hermite polynomials. For a statistical ensemble of identical oscillators defined by the Yurke–Buks energy operator, we calculate the partition function, average energy, free energy and entropy. Using the diagonal element of the canonical density matrix of this ensemble in the coordinate representation, we define a probability density, which appears to be a deformed Gaussian distribution. A peculiarity of this probability density is that it may reveal, when plotted as a function of the position variable, a shape with two peaks located symmetrically with respect to the central point.

The Norm of the Non-Self-Adjoint Harmonic Oscillator Semigroup

We identify the norm of the semigroup generated by the non-self-adjoint harmonic oscillator acting on $L^2(\Bbb{R})$, for all complex times where it is bounded. We relate this problem to embeddings between Gaussian-weighted spaces of holomorphic functions, and we show that the same technique applies, in any dimension, to the semigroup $e^{-tQ}$ generated by an elliptic quadratic operator acting on $L^2(\Bbb{R}^n)$. The method used --- identifying the exponents of sharp products of Mehler formulas --- is elementary and is inspired by more general works of L. H\"ormander, A. Melin, and J. Sj\"ostrand.

Propagators of isochronous an-harmonic oscillators and Mehler formula for the exceptional Hermite polynomials

It is shown that fundamental solutions Kσ(x,y;t)=〈x|e −i Hσt|y〉 of the non-stationary Schrödinger equation (Green functions, or propagators) for the rational extensions of the Harmonic oscillator Hσ=Hosc+ΔVσ are expressed in terms of elementary functions only. An algorithm to calculate explicitly Kσ for an arbitrary increasing sequence of positive integers σ is given, and compact expressions for K{1,2} and K{2,3} are presented. A generalization of Mehler’s formula to the case of exceptional Hermite polynomials is given.

Regular transformation groups based on Fourier-Gauss transforms

We discuss different representations of the white noise test functions (E)β , 0 ≤ β < 1 by introducing generalized Wick tensors. As an application we state the following generalization of the Mehler formula for the Ornstein-Uhlenbeck semigroup, see [7, p. 237]: Let a, b ∈ C, 0 ≤ β < 1. Then a · ∆G + b · N is the infinitesimal generator of the following regular differentiable one parameter transformation group { Pa,b,t } t∈R ⊂ GL((E)β): (i) if b = 0 then for all φ ∈ (E)β , t ∈ R :