Mehler's Formula

The Mehler kernel is a complex-valued function found to be the propagator of the quantum harmonic oscillator.

Mehler's formula

defined a function

and showed, in modernized notation, that it can be expanded in terms of Hermite polynomials (.) based on weight function exp(−²) as
:$E(x,y) = \sum_^\infty \frac ~ \mathit_n(x)\mathit_n(y) ~.$

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 to
:$\frac = \frac-x^2\varphi \equiv D_x \varphi ~.$

The orthonormal eigenfunctions of the operator are the Hermite functions,
:$\psi_n = \frac,$
with corresponding eigenvalues (2+1), furnishing particular solutions
: $\varphi_n(x, t)= e^ ~H_n(x) \exp(-x^2/2) ~.$

The general solution is then a linear combination of these; when fitted to the initial condition , the general solution reduces to
: $\varphi(x,t)= \int K(x,y;t) \varphi(y,0) dy ~,$
where the kernel has the separable representation
:$K(x,y;t)\equiv\sum_ \frac ~ H_n(x) H_n(y) \exp(-(x^2+y^2)/2)~.$

Utilizing Mehler's formula then yields
:$~.$

On substituting this in the expression for with the value for , Mehler's kernel finally reads

When = 0, variables and 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 .

When using the usual physics conventions of defining the quantum harmonic oscillator instead via
:$i \frac = \frac\left(-\frac+x^2\right) \varphi \equiv H \varphi,$
and assuming natural length and energy scales, then the Mehler kernel becomes the Feynman propagator $K_$ which reads
:$\langle x \mid \exp (-itH) \mid y \rangle \equiv K_(x,y;t)= \frac \exp \left(\frac\left ((x^2+y^2)\cos t - 2xy\right )\right ),\quad t< \pi,$
i.e. $K_(x,y;t) = K(x,y; i t/2 ).$

When $t>\pi$ the $i \sin t$ in the inverse square-root should be replaced by $, \sin t,$ and $K_$ should be
multiplied by an extra Maslov phase factor
:$\exp\left(i\theta_\right) = \exp\left(-i\frac\left(\frac +\left\lfloor\frac\right\rfloor \right)\right).$

When $t = \pi/2$ the general solution is proportional to the Fourier transform $\mathcal$ of the initial conditions $\varphi_0(y)\equiv\varphi(y,0)$ since
: \varphi(x, t=\pi/2) = \int K_(x,y; \pi/2) \varphi(y,0) dy = \frac \int \exp(-i x y) \varphi(y,0) dy = \exp(-i \pi /4) \mathcal varphi_0x) ~,
and the exact Fourier transform is thus obtained from the quantum harmonic oscillator's number operator written as
:$N \equiv \frac\left(x-\frac\right)\left(x+\frac\right) = H-\frac = \frac\left(-\frac+x^2-1\right) ~$
since the resulting kernel
:$\langle x \mid \exp (-it N) \mid y \rangle \equiv K_(x,y;t) = \exp(i t /2) K_(x,y; t) = \exp(i t /2) K(x,y;i t /2)$
also compensates for the phase factor still arising in $K_$ and $K$, i.e.
:\varphi(x,t = \pi/2)= \int K_(x,y; \pi/2) \varphi(y,0) dy = \mathcal varphi_0x)~,
which shows that the number operator can be interpreted via the Mehler kernel as the generator of fractional Fourier transforms for arbitrary values of , and of the conventional Fourier transform $\mathcal$ for the particular value $t = \pi/2$, with the Mehler kernel providing an active transform, while the corresponding passive transform is already embedded in the basis change from position to momentum space. The eigenfunctions of $N$ are still the Hermite functions $\psi_n(x)$ which are therefore also Eigenfunctions of $\mathcal$.

Probability version

The result of Mehler can also be linked to probability. For this, the variables should be rescaled as , , so as to change from the 'physicist's' Hermite polynomials (.) (with weight function exp(−²)) to "probabilist's" Hermite polynomials (.) (with weight function exp(−²/2)). Then, becomes
:$\frac 1\exp\left(-\frac\right) = \sum_^\infty \frac ~ \mathit_n(x)\mathit_n(y) ~.$

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 having zero means and unit variances:
:$p(x,y) = \frac 1\exp\left(-\frac\right) ~,$
and are the corresponding probability densities of and (both standard normal).

There follows the usually quoted form of the result (Kibble 1945)
:$p(x,y) = p(x) p(y)\sum_^\infty \frac ~ \mathit_n(x)\mathit_n(y) ~.$

This expansion is most easily derived by using the two-dimensional Fourier transform of , 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
:$\exp( -(u_1^2 + u_2^2)/2 ) \sum_^\infty \frac (u_1 u_2)^n ~.$
The Inverse Fourier transform then immediately yields the above expansion formula.

This result can be extended to the multidimensional case.

Fractional Fourier transform

Since Hermite functions are orthonormal eigenfunctions of the Fourier transform,
:\mathcal psi_ny)=(-i)^n \psi_n(y) ~,
in harmonic analysis and signal processing, they diagonalize the Fourier operator,
:\mathcal y) =\int dx f(x) \sum_ (-i)^n \psi_n(x) \psi_n(y) ~.

Thus, the continuous generalization for real angle can be readily defined ( Wiener, 1929; Condon, 1937 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
/ref>), the fractional Fourier transform (FrFT), with kernel
:$\mathcal_\alpha = \sum_ (-i)^ \psi_n(x) \psi_n(y) ~.$

This is a ''continuous family of linear transforms generalizing the Fourier transform'', such that, for , it reduces to the standard Fourier transform, and for to the inverse Fourier transform.

The Mehler formula, for = exp(−i), thus directly provides
:\mathcal_\alpha y) =
\sqrt ~ e^
\int_^\infty
e^ f(x)\, \mathrmx ~.

The square root is defined such that the argument of the result lies in the interval ��''π'' /2, ''π'' /2

If is an integer multiple of , then the above cotangent and cosecant functions diverge. In the limit, the kernel goes to a Dirac delta function in the integrand, or , for an even or odd multiple of , respectively. Since $\mathcal^2$ = (−), $\mathcal_\alpha$ must be simply or for an even or odd multiple of , respectively.

See also

*
* Heat kernel
* Hermite polynomials
* Parabolic cylinder functions
*

References

* Nicole Berline, Ezra Getzler, and Michèle Vergne (2013). ''Heat Kernels and Dirac Operators'', (Springer: Grundlehren Text Editions) Paperback
* {{Cite journal, author=Louck, J. D., journal=Advances in Applied Mathematics,
volume =2 , date= 1981, pages= 239–249, title=Extension of the Kibble-Slepian formula for Hermite polynomials using boson operator methods , issue=3, doi = 10.1016/0196-8858(81)90005-1, doi-access=free
* H. M. Srivastava and J. P. Singhal (1972). "Some extensions of the Mehler formula", ''Proc. Amer. Math. Soc.'' 31: 135–141.
online

Parabolic partial differential equations
Orthogonal polynomials
Mathematical physics
Multivariate continuous distributions