Mehler kernel

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Mehler's formula[edit]

Mehler (1866) defined a function[1]

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

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[edit]

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

\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

 \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

\varphi(x,t)= \int K(x,y;t) \varphi(y,0) dy ~,

where the kernel K has the separable representation

K(x,y;t)\equiv\sum_{n\ge 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{\sum_{n\ge 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

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,

K(x,y;0)= \delta(x-y)~.

As a fundamental solution, the kernel is additive,

\int dy K(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[edit]

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

 
\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:

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]

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

 c(iu_1, iu_2) = \exp (- (u_1^2 + u_2^2 - 2 \rho u_1 u_2)/2)~.

This may be expanded as

 \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[edit]

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

\mathcal{F} [\psi_n](y)=(-i)^n \psi_n(y) ~,

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

\mathcal{F}[f](y) =\int dx f(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

\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

\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)~ y x - \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 \mathcal{F}^2[f ] = f(−x), \mathcal{F}_\alpha[f ] must be simply f(x) or f(−x) for α an even or odd multiple of π, respectively.

Also see[edit]

References[edit]

  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,
     (x,y) {\mathbf M} \begin{pmatrix}{x}\\{y}\end{pmatrix} ~,~   where
     {\mathbf M} \equiv\text{cosech} (2t) \begin{pmatrix} \cosh (2t) &-1\\-1&\cosh (2t)\end{pmatrix} ~,
    so it preserves the symplectic metric,
     {\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. Nat. 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)