# Mehler kernel

## Mehler's formula

Mehler (1866) defined a function[1]

 ${\displaystyle E(x,y)={\frac {1}{\sqrt {1-\rho ^{2}}}}\exp \left(-{\frac {\rho ^{2}(x^{2}+y^{2})-2\rho xy}{(1-\rho ^{2})}}\right)~,}$

and showed, in modernized notation,[2] that it can be expanded in terms of Hermite polynomials H(.) based on weight function exp(−x²) as

${\displaystyle E(x,y)=\sum _{n=0}^{\infty }{\frac {(\rho /2)^{n}}{n!}}~{\mathit {H}}_{n}(x){\mathit {H}}_{n}(y)~.}$

This result is useful, in modified form, in quantum physics, probability theory, and harmonic analysis.

## Physics version

In physics, the fundamental solution, (Green's function), or propagator of the Hamiltonian for the quantum harmonic oscillator is called the Mehler kernel. It provides the fundamental solution---the most general solution[3] φ(x,t) to

${\displaystyle {\frac {\partial \varphi }{\partial t}}={\frac {\partial ^{2}\varphi }{\partial x^{2}}}-x^{2}\varphi \equiv D_{x}\varphi ~.}$

The orthonormal eigenfunctions of the operator D are the Hermite functions, ψn = Hn(x) exp(−x²/2) /√2n n! √π , with corresponding eigenvalues (2n+1), furnishing particular solutions

${\displaystyle \varphi _{n}(x,t)=e^{-(2n+1)t}~H_{n}(x)\exp(-x^{2}/2)~.}$

The general solution is then a linear combination of these; when fitted to the initial condition φ(x,0), the general solution reduces to

${\displaystyle \varphi (x,t)=\int K(x,y;t)\varphi (y,0)dy~,}$

where the kernel K has the separable representation

${\displaystyle K(x,y;t)\equiv \sum _{n\geq 0}{\frac {e^{-(2n+1)t}}{{\sqrt {\pi }}2^{n}n!}}~H_{n}(x)H_{n}(y)\exp(-(x^{2}+y^{2})/2)~.}$

Utilizing Mehler's formula then yields

${\displaystyle \displaystyle {\sum _{n\geq 0}{\frac {(\rho /2)^{n}}{n!}}H_{n}(x)H_{n}(y)\exp(-(x^{2}+y^{2})/2)={1 \over {\sqrt {(1-\rho ^{2})}}}\exp {4xy\rho -(1+\rho ^{2})(x^{2}+y^{2}) \over 2(1-\rho ^{2})}}~.}$

On substituting this in the expression for K with the value exp(−2t) for ρ, Mehler's kernel finally reads

 ${\displaystyle K(x,y;t)={\frac {1}{\sqrt {2\pi \sinh(2t)}}}~\exp {\Bigl (}-\coth(2t)~(x^{2}+y^{2})/2+{\text{cosech}}(2t)~xy{\Bigr )}.}$

When t = 0, variables x and y coincide, resulting in the limiting formula necessary by the initial condition,

${\displaystyle K(x,y;0)=\delta (x-y)~.}$

As a fundamental solution, the kernel is additive,

${\displaystyle \int dyK(x,y;t)K(y,z;t')=K(x,z;t+t')~.}$

This is further related to the symplectic rotation structure of the kernel K.[4]

## Probability version

The result of Mehler can also be linked to probability. For this, the variables should be rescaled as xx/√2, y → y/√2, so as to change from the 'physicist's' Hermite polynomials H(.) (with weight function exp(−x²)) to "probabilist's" Hermite polynomials He(.) (with weight function exp(−x²/2)). Then, E becomes

${\displaystyle {\frac {1}{\sqrt {1-\rho ^{2}}}}\exp \left(-{\frac {\rho ^{2}(x^{2}+y^{2})-2\rho xy}{2(1-\rho ^{2})}}\right)=\sum _{n=0}^{\infty }{\frac {\rho ^{n}}{n!}}~{\mathit {He}}_{n}(x){\mathit {He}}_{n}(y)~.}$

The left-hand side here is p(x,y)/p(x)p(y) where p(x,y) is the bivariate Gaussian probability density function for variables x,y having zero means and unit variances:

${\displaystyle p(x,y)={\frac {1}{2\pi {\sqrt {1-\rho ^{2}}}}}\exp \left(-{\frac {(x^{2}+y^{2})-2\rho xy}{2(1-\rho ^{2})}}\right)~,}$

and p(x), p(y) are the corresponding probability densities of x and y.

There follows the usually quoted form of the result (Kibble 1945)[5]

${\displaystyle p(x,y)=p(x)p(y)\sum _{n=0}^{\infty }{\frac {\rho ^{n}}{n!}}~{\mathit {He}}_{n}(x){\mathit {He}}_{n}(y)~.}$

This expansion is most easily derived by using the two-dimensional Fourier transform of p(x,y), which is

${\displaystyle c(iu_{1},iu_{2})=\exp(-(u_{1}^{2}+u_{2}^{2}-2\rho u_{1}u_{2})/2)~.}$

This may be expanded as

${\displaystyle \exp(-(u_{1}^{2}+u_{2}^{2})/2)\sum _{n=0}^{\infty }{\frac {\rho ^{n}}{n!}}(u_{1}u_{2})^{n}~.}$

The Inverse Fourier transform then immediately yields the above expansion formula.

This result can be extended to the multidimensional case (Kibble 1945, Slepian 1972,[6] Hörmander 1985 [7]).

## Fractional Fourier transform

Since Hermite functions ψn are orthonormal eigenfunctions of the Fourier transform,

${\displaystyle {\mathcal {F}}[\psi _{n}](y)=(-i)^{n}\psi _{n}(y)~,}$

in harmonic analysis and signal processing, they diagonalize the Fourier operator,

${\displaystyle {\mathcal {F}}[f](y)=\int dxf(x)\sum _{n\geq 0}(-i)^{n}\psi _{n}(x)\psi _{n}(y)~.}$

Thus, the continuous generalization for real angle α can be readily defined (Wiener, 1929;[8] Condon, 1937[9]), the fractional Fourier transform (FrFT), with kernel

${\displaystyle {\mathcal {F}}_{\alpha }=\sum _{n\geq 0}(-i)^{2\alpha n/\pi }\psi _{n}(x)\psi _{n}(y)~.}$

This is a continuous family of linear transforms generalizing the Fourier transform, such that, for α = π/2, it reduces to the standard Fourier transform, and for α = −π/2 to the inverse Fourier transform.

The Mehler formula, for ρ = exp(−iα), thus directly provides

${\displaystyle {\mathcal {F}}_{\alpha }[f](y)={\sqrt {\frac {1-i\cot(\alpha )}{2\pi }}}~e^{i{\frac {\cot(\alpha )}{2}}y^{2}}\int _{-\infty }^{\infty }e^{-i\left(\csc(\alpha )~yx-{\frac {\cot(\alpha )}{2}}x^{2}\right)}f(x)\,\mathrm {d} x~.}$

The square root is defined such that the argument of the result lies in the interval [−π /2, π /2].

If α is an integer multiple of π, then the above cotangent and cosecant functions diverge. In the limit, the kernel goes to a Dirac delta function in the integrand, δ(x−y) or δ(x+y), for α an even or odd multiple of π, respectively. Since ${\displaystyle {\mathcal {F}}^{2}}$[f ] = f(−x), ${\displaystyle {\mathcal {F}}_{\alpha }}$[f ] must be simply f(x) or f(−x) for α an even or odd multiple of π, respectively.

## References

1. ^ Mehler, F. G. (1866), "Ueber die Entwicklung einer Function von beliebig vielen Variabeln nach Laplaceschen Functionen höherer Ordnung", Journal für Reine und Angewandte Mathematik (in German) (66): 161–176, ISSN 0075-4102, JFM 066.1720cj (cf. p 174, eqn (18) & p 173, eqn (13) )
2. ^ Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1955), Higher transcendental functions. Vol. II, McGraw-Hill (scan:   p.194 10.13 (22))
3. ^ Pauli, W., Wave Mechanics: Volume 5 of Pauli Lectures on Physics (Dover Books on Physics, 2000) ISBN 0486414620 ; See section 44.
4. ^ The quadratic form in its exponent, up to a factor of −1/2, involves the simplest (unimodular, symmetric) symplectic matrix in Sp(2,ℝ). That is,
${\displaystyle (x,y){\mathbf {M} }{\begin{pmatrix}{x}\\{y}\end{pmatrix}}~,~}$   where
${\displaystyle {\mathbf {M} }\equiv {\text{cosech}}(2t){\begin{pmatrix}\cosh(2t)&-1\\-1&\cosh(2t)\end{pmatrix}}~,}$
so it preserves the symplectic metric,
${\displaystyle {\mathbf {M} }^{\text{T}}~{\begin{pmatrix}0&1\\-1&0\end{pmatrix}}~{\mathbf {M} }={\begin{pmatrix}0&1\\-1&0\end{pmatrix}}~.}$
5. ^ Kibble, W. F. (1945), "An extension of a theorem of Mehler's on Hermite polynomials", Proc. Cambridge Philos. Soc. 41: 12–15, doi:10.1017/S0305004100022313, MR 0012728
6. ^ Slepian, David (1972), "On the symmetrized Kronecker power of a matrix and extensions of Mehler's formula for Hermite polynomials", SIAM Journal on Mathematical Analysis 3: 606–616, doi:10.1137/0503060, ISSN 0036-1410, MR 0315173
7. ^ Hörmander, Lars (1995). "Symplectic classification of quadratic forms, and general Mehler formulas". Mathematische Zeitschrift 219: 413–449. doi:10.1007/BF02572374.
8. ^ Wiener, N (1929), "Hermitian Polynomials and Fourier Analysis", Journal of Mathematics and Physics 8: 70-73.
9. ^ Condon, E. U. (1937). "Immersion of the Fourier transform in a continuous group of functional transformations", Proc. Natl. Acad. Sci. USA 23, 158–164. online
• Nicole Berline, Ezra Getzler, and Michèle Vergne (2013). Heat Kernels and Dirac Operators, (Springer: Grundlehren Text Editions) Paperback ISBN 3540200622
• Louck, J. D. (1981). "Extension of the Kibble-Slepian formula for Hermite polynomials using boson operator methods". Advances in Applied Mathematics 2: 239–249. doi:10.1016/0196-8858(81)90005-1.
• H. M. Srivastava and J. P. Singhal (1972). "Some extensions of the Mehler formula", Proc. Amer. Math. Soc. 31: 135-141. (online)