Mehler kernel

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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, ERAM 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)