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Mehler kernel

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Mehler's formula

Mehler (1866) defined a function

Contents

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

E ( x , y ) = ∑ n = 0 ∞ ( ρ / 2 ) n n !   H n ( x ) 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 φ(x,t) to

∂ φ ∂ t = ∂ 2 φ ∂ x 2 − x 2 φ ≡ D x φ   .

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

φ n ( x , t ) = e − ( 2 n + 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

φ ( x , t ) = ∫ K ( x , y ; t ) φ ( y , 0 ) d y   ,

where the kernel K has the separable representation

K ( x , y ; t ) ≡ ∑ n ≥ 0 e − ( 2 n + 1 ) t π 2 n n !   H n ( x ) H n ( y ) exp ⁡ ( − ( x 2 + y 2 ) / 2 )   .

Utilizing Mehler's formula then yields

∑ n ≥ 0 ( ρ / 2 ) n n ! H n ( x ) H n ( y ) exp ⁡ ( − ( x 2 + y 2 ) / 2 ) = 1 ( 1 − ρ 2 ) exp ⁡ 4 x y ρ − ( 1 + ρ 2 ) ( x 2 + y 2 ) 2 ( 1 − ρ 2 )   .

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

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

As a fundamental solution, the kernel is additive,

∫ d y 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.

Probability version

The result of Mehler can also be linked to probability. For this, the variables should be rescaled as x → x/√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

1 1 − ρ 2 exp ⁡ ( − ρ 2 ( x 2 + y 2 ) − 2 ρ x y 2 ( 1 − ρ 2 ) ) = ∑ n = 0 ∞ ρ n n !   H e n ( x ) H e 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 ) = 1 2 π 1 − ρ 2 exp ⁡ ( − ( x 2 + y 2 ) − 2 ρ x y 2 ( 1 − ρ 2 ) )   ,

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)

p ( x , y ) = p ( x ) p ( y ) ∑ n = 0 ∞ ρ n n !   H e n ( x ) H e n ( y )   .

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

c ( i u 1 , i u 2 ) = exp ⁡ ( − ( u 1 2 + u 2 2 − 2 ρ u 1 u 2 ) / 2 )   .

This may be expanded as

exp ⁡ ( − ( u 1 2 + u 2 2 ) / 2 ) ∑ n = 0 ∞ ρ 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, Hörmander 1985 ).

Fractional Fourier transform

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

F [ ψ n ] ( y ) = ( − i ) n ψ n ( y )   ,

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

F [ f ] ( y ) = ∫ d x f ( x ) ∑ n ≥ 0 ( − i ) n ψ n ( x ) ψ n ( y )   .

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

F α = ∑ n ≥ 0 ( − i ) 2 α n / π ψ n ( x ) ψ 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

F α [ f ] ( y ) = 1 − i cot ⁡ ( α ) 2 π   e i cot ⁡ ( α ) 2 y 2 ∫ − ∞ ∞ e − i ( csc ⁡ ( α )   y x − cot ⁡ ( α ) 2 x 2 ) f ( x ) 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 F 2 [f ] = f(−x), F α [f ] must be simply f(x) or f(−x) for α an even or odd multiple of π, respectively.

References

Mehler kernel Wikipedia
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