Mehler Formula Research Articles

Opérateurs linéaires gaussiens

We extend operators from the Cameron-Martin space to Gaussian Lusinian locally convex space. We then are allowed to give sense to the Mehler formula for every such bounded operator. An application is made to Hilbert-Schmidt operators. Next we show that capacities asociated to second quantizations of operators are tight on compact sets, and this is a general result even if the underlying space is not a Banach space.

Laplace asymptotic expansions of conditional Wiener integrals and generalised Mehler kernel formulae for Hamiltonians on L2(Rn)

Laplace asymptotic expansions of conditional Wiener integrals and generalised Mehler kernel formulae for Hamiltonians on L2(Rn)

Density and Correlations in Large One and Three Dimensional Systems of Fermions

The density and correlations are studied for a system of fermions moving in an average harmonic oscillator potential in one and three dimensions when the number of fermions are large. The technique is based on the use of Mehler's formula for the product of two Hermite functions and con-. tour integrations. Wigner transform is then calculated using this technique. It is shown that it becomes a step function when number of fermions become large. By integrating out momenta, an expression for the density of fermions is derived. Two-particle correlation function is studied in detail. Dyson's type of correlations for a system of nucleons is studied. The connection between one dimensional system of fermions and eigenvalues of a random Hermitean Hamiltonian is shown. The Coulomb energy difference of isobars is studied using nucleon density and correlations.

The Combinatorics of q-Hermite polynomials and the Askey—Wilson Integral

The q-Hermite polynomials are defined as a q -analogue of the matching polynomial of a complete graph. This allows a combinatorial evaluation of the integral used to prove the orthogonality of Askey and Wilson's 4 φ 3 polynomials. A special case of this result gives the linearization formula for q -Hermite polynomials. The moments and associated continued fraction are explicitly given. Another set of polynomials, closely related to the q -Hermite, is defined. These polynomials have a combinatorial interpretation in terms of finite vector spaces which give another proof of the linearization formula and the q -analogue of Mehler's formula.

Laplace asymptotic expansions of conditional Wiener integrals and generalised Mehler kernel formulae for Hamiltonians on L2(Rn)

The authors extend their previous results (1982) to derive Laplace asymptotic expansions for multi-dimensional conditional Wiener integrals. Applications are given to obtaining generalised Mehler kernel formulae, up to arbitrarily high orders in h(cross), for the kernels exp(-TH(h(cross))/h(cross))(x,y) and exp(-TH(h(cross),iB)/h(cross))(x,y) for T>0, where H(h(cross))=(-h(cross)2/2) Delta 3+V(x) and H(h(cross),B)=2-1(ih(cross) Del +2-1(B V-product x))2+V(x). Here V in Cinfinity (R3) is a real-valued potential which is bounded below together with its Hessian matrix and B is a constant vector. They emphasise that these results are valid for non-separable systems.

Number of energy levels outside Wigner's semicircle

The total probability outside Wigner's (1962) semicircle is calculated using Mehler's formula for the Hermite polynomials. In the limit N to infinity , N being the dimension of the random Gaussian Hermitian matrix it is shown that this probability is 0.05N-32/. Some implications of this small probability are discussed.

Laplace asymptotic expansions of conditional Wiener integrals and generalized Mehler kernel formulas

Imitating Schilder’s results for Wiener integrals rigorous Laplace asymptotic expansions are proven for conditional Wiener integrals. Applications are given for deriving generalized Mehler kernel formulas, up to arbitrarily high orders in powers of ℏ, for exp{−TH(ℏ)/ℏ}(x, y), T>0 where H(ℏ)=[(−ℏ2/2)Δ1+V], Δ1 being the one-dimensional Laplacian, V being a real-valued potential V∈C∞(R), bounded below, together with its second derivative.

Extension of the Kibble-Slepian formula for Hermite polynomials using boson operator methods

The Kibble-Slepian formula expresses the exponential of a quadratic form Q( x) = x t S( I + S) −1 x, S t = S, in n variables x = col( x 1,…, x n ) as a series of products of Hermite polynomials, thus generalizing Mehler's formula. This extension is restricted, however, to the case where the diagonal elements of the symmetric matrix S are all unity. We derive the general formula for an arbitrary symmetric matrix S, where I + S is positive definite, using techniques familiar from the boson operator treatment of the harmonic oscillator in quantum mechanics.

Some Hermite polynomial identities and their combinatorics

The multilinear extensions of the Mehler formula found by Kibble, Slepian and Louck are shown to be equivalent. They can all be proved by using the combinatorial set-up of involutionary graphs, and so thus the classical Doetsch identity.

Dilations and Mehler's kernel

An explicit formula for the second quantization of a contraction matrix on a finite dimensional Hilbert space is derived by means of dilation theory. When the dimension is one, this reduces to Mehler's formula.

A combinatorial proof of the Mehler formula

A combinatorial proof of the Mehler formula on Hermite polynomials is given that is based upon the techniques of the partitional complex.

A modified Mehler formula and the Green function of the free electron in a uniform magnetic field

A closed representation is obtained for the Green function of the stationary Schrodinger equation of a free electron in a uniform magnetic field.

The Mehler formula and the Green function of the multi-dimensional isotropic harmonic oscillator

Using a generalized Mehler formula, a closed representation is obtained for the Green function of the stationary Schrodinger equation for a multidimensional isotropic harmonic oscillator.

A new proof of a Paley—Wiener type theorem for the Jacobi transform

which generalizes the Mehler-Fok transform, was studied by Titchmarsh [23, w 17], Olevskii [21], Braaksma and Meulenbeld [2], Flensted--Jensen [9], [11, w and w and Flensted--Jensen and Koornwinder [12]. Some papers by Ch6bli [3], [4], [5] deal with a larger class of integral transforms which includes the Jacobi transform. An even more general class was considered by Braaksma and De Shoo [24]. In the present paper short proofs will be given of a Paley--Wiener type theorem and the inversion formula for the Jacobi transform. The L2-theory, i.e. the Plancherel theorem, is then an easy consequence. These results were earlier obtained by Flensted--Jensen [9], [11, w and by Ch6bli [5]. However, to prove the Paley-Wiener theorem these two authors needed the L2-theory, which can be obtained as a corollary of the Weyl S t o n e T i t c h m a r s h K o d a i r a theorem about the spectral decomposition of a singular Sturm--Liouville operator (cf. for instance Dunford and Schwartz [6, Chap. 13, w The proofs presented here exploit the properties of Jacobi functions as hypergeometric functions and no general theorem needs to be invoked. Furthermore, it turns out that the Paley--Wiener theorem, which was proved by Flensted---Jensen [11, w for real c~, fi, ~ > 1, holds for all complex values of ~ and ft. The key formula in this paper is a generalized Mehler formula

Some generalizations of Mehler's formula

Some generalizations of Mehler's formula

Some extensions of the Mehler formula, II

Some extensions of the Mehler formula, II

Some extensions of the Mehler formula

By using certain operational techniques, the authors prove an elegant unification of several extensions of the well-known Mehler formula for Hermite polynomials, given recently by L. Carlitz([

Some formulas involving the products of several Jacobi or Laguerre polynomials

By using the principle of mathematical induction and certain operational techniques, the authors prove a number of multiple series relations associated with the products of several Jacobi or Laguerre polynomials. The results obtained here are analogous to the recent extensions of the well-known Mehler formula for Hermite polynomials, given by Carlitz and the authors.

Some Extensions of the Mehler Formula

Some Extensions of the Mehler Formula

Spectral Contours of Continuous Absorption Bands of Complex Molecules

An analytical expression for the description of the shape of continuous absorption bands is calculated within the framework of a new one-dimensional potential model. The calculations are based on a semi-classical approximation of the Franck—Condon factors and on the application of the Mehler formula for the sum calculation of statistically weighted position probability densities.