Mehler kernel

The Mehler kernel is a complex-valued function found to be the propagator of the quantum harmonic oscillator.
Mehler's formula.
 defined a function
\exp\left(-\frac{\rho^2 (x^2+y^2)- 2\rho xy}{(1-\rho^2)}\right)~, </math>
and showed, in modernized notation, that it can be expanded in terms of Hermite polynomials (.) based on weight function exp(−²) as
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 of the Hamiltonian for the quantum harmonic oscillator is called the Mehler kernel. It provides the fundamental solution---the most general solution to 
The orthonormal eigenfunctions of the operator are the Hermite functions,
with corresponding eigenvalues (2+1), furnishing particular solutions
The general solution is then a linear combination of these; when fitted to the initial condition , the general solution reduces to
where the kernel has the separable representation
Utilizing Mehler's formula then yields 
On substituting this in the expression for with the value exp(−2) for , Mehler's kernel finally reads
~\exp\Bigl(-\coth(2t)~(x^2+y^2)/2 + \text{cosech}(2t)~xy\Bigr).</math>
When = 0, variables and coincide, resulting in the limiting formula necessary by the initial condition,
As a fundamental solution, the kernel is additive,
This is further related to the symplectic rotation structure of the kernel .
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
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: 
and are the corresponding probability densities of and (both standard normal).
There follows the usually quoted form of the result (Kibble 1945)
This expansion is most easily derived by using the two-dimensional Fourier transform of , which is
This may be expanded as
The Inverse Fourier transform then immediately yields the above expansion formula.
This result can be extended to the multidimensional case (Kibble 1945, Slepian 1972, Hörmander 1985 ).
Fractional Fourier transform.
Since Hermite functions are orthonormal eigenfunctions of the Fourier transform,
in harmonic analysis and signal processing, they diagonalize the Fourier operator,
Thus, the continuous generalization for real angle can be readily defined (Wiener, 1929; Condon, 1937), the fractional Fourier transform (FrFT), with kernel
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
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 formula_19[ ] = (−), formula_20[ ] must be simply or for an even or odd multiple of , respectively.