Mehler kernel

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

Mehler's formula

Template:Harvs defined a function

$E(x,y)=\frac{1}{\sqrt{1-{\rho }^2}}\exp (-\frac{{\rho }^2(x^2+y^2)-2\rho xy}{(1-{\rho }^2)}),$

and showed, in modernized notation, that it can be expanded in terms of Hermite polynomials Template:Mvar(.) based on weight function exp(−Template:Mvar²) as

$E(x,y)=\sum _{n=0}^{\mathrm{\infty }}\frac{(\rho /2{)}^n}{n!}{𝐻}_n(x){𝐻}_n(y).$

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 Template:Math to

$\frac{\partial \phi }{\partial t}=\frac{{\partial }^2\phi }{\partial x^2}-x^2\phi \equiv D_x\phi .$

The orthonormal eigenfunctions of the operator Template:Mvar are the Hermite functions,

${\psi }_n=\frac{H_n(x)\exp (-x^2/2)}{\sqrt{2^{n}n!\sqrt{\pi }}},$

with corresponding eigenvalues (-2Template:Mvar-1), furnishing particular solutions

${\phi }_n(x,t)=e^{-(2n+1)t}{H}_n(x)\exp (-x^2/2).$

The general solution is then a linear combination of these; when fitted to the initial condition Template:Math, the general solution reduces to

$\phi (x,t)=\int K(x,y;t)\phi (y,0)dy,$

where the kernel Template:Mvar 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

$\sum _{n\ge 0}\frac{(\rho /2{)}^n}{n!}{H}_n(x)H_n(y)\exp (-(x^2+y^2)/2)=\frac{1}{\sqrt{(1-{\rho }^2)}}\exp \left(\frac{4xy\rho -(1+{\rho }^2)(x^2+y^2)}{2(1-{\rho }^2)}\right).$

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

$K(x,y;t)=\frac{1}{\sqrt{2\pi \sinh (2t)}}\exp (-\coth (2t)(x^2+y^2)/2+\mathrm{csch}(2t)xy).$

When Template:Mvar = 0, variables Template:Mvar and Template:Mvar 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,

$\int dyK(x,y;t)K(y,z;t^{\prime })=K(x,z;t+t^{\prime }).$

This is further related to the symplectic rotation structure of the kernel Template:Mvar.

When using the usual physics conventions of defining the quantum harmonic oscillator instead via

$i\frac{\partial \phi }{\partial t}=\frac{1}{2}(-\frac{{\partial }^2}{\partial x^2}+x^2)\phi \equiv H\phi ,$

and assuming natural length and energy scales, then the Mehler kernel becomes the Feynman propagator $K_H$ which reads

$⟨x\mid \exp (-itH)\mid y⟩\equiv K_H(x,y;t)=\frac{1}{\sqrt{2\pi i\sin t}}\exp \left(\frac{i}{2\sin t}((x^2+y^2)\cos t-2xy)\right),t<\pi ,$

i.e. $K_H(x,y;t)=K(x,y;it/2).$

When $t>\pi$ the $i\sin t$ in the inverse square-root should be replaced by $|\sin t|$ and $K_H$ should be multiplied by an extra Maslov phase factor

$\exp \left(i{\theta }_{\mathrm{M}\mathrm{a}\mathrm{s}\mathrm{l}\mathrm{o}\mathrm{v}}\right)=\exp (-i\frac{\pi }{2}(\frac{1}{2}+\lfloor \frac{t}{\pi }\rfloor )).$

When $t=\pi /2$ the general solution is proportional to the Fourier transform $ℱ$ of the initial conditions ${\phi }_0(y)\equiv \phi (y,0)$ since

$\phi (x,t=\pi /2)=\int K_H(x,y;\pi /2)\phi (y,0)dy=\frac{1}{\sqrt{2\pi i}}\int \exp (-ixy)\phi (y,0)dy=\exp (-i\pi /4)ℱ[{\phi }_0](x),$

and the exact Fourier transform is thus obtained from the quantum harmonic oscillator's number operator written as

$N\equiv \frac{1}{2}(x-\frac{\partial }{\partial x})(x+\frac{\partial }{\partial x})=H-\frac{1}{2}=\frac{1}{2}(-\frac{{\partial }^2}{\partial x^2}+x^2-1)$

since the resulting kernel

$⟨x\mid \exp (-itN)\mid y⟩\equiv K_N(x,y;t)=\exp (it/2)K_H(x,y;t)=\exp (it/2)K(x,y;it/2)$

also compensates for the phase factor still arising in $K_H$ and $K$, i.e.

$\phi (x,t=\pi /2)=\int K_N(x,y;\pi /2)\phi (y,0)dy=ℱ[{\phi }_0](x),$

which shows that the number operator can be interpreted via the Mehler kernel as the generator of fractional Fourier transforms for arbitrary values of Template:Mvar, and of the conventional Fourier transform $ℱ$ 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 the usual Hermite functions ${\psi }_n(x)$ which are therefore also Eigenfunctions of $ℱ$.

Probability version

The result of Mehler can also be linked to probability. For this, the variables should be rescaled as Template:Math, Template:Math, so as to change from the 'physicist's' Hermite polynomials Template:Mvar(.) (with weight function exp(−Template:Mvar²)) to "probabilist's" Hermite polynomials Template:Math(.) (with weight function exp(−Template:Mvar²/2)). Then, Template:Mvar becomes

$\frac{1}{\sqrt{1-{\rho }^2}}\exp (-\frac{{\rho }^2(x^2+y^2)-2\rho xy}{2(1-{\rho }^2)})=\sum _{n=0}^{\mathrm{\infty }}\frac{{\rho }^n}{n!}{𝐻𝑒}_n(x){𝐻𝑒}_n(y).$

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 Template:Math having zero means and unit variances:

$p(x,y)=\frac{1}{2\pi \sqrt{1-{\rho }^2}}\exp (-\frac{(x^2+y^2)-2\rho xy}{2(1-{\rho }^2)}),$

and Template:Math are the corresponding probability densities of Template:Mvar and Template:Mvar (both standard normal).

There follows the usually quoted form of the result (Kibble 1945)

$p(x,y)=p(x)p(y)\sum _{n=0}^{\mathrm{\infty }}\frac{{\rho }^n}{n!}{𝐻𝑒}_n(x){𝐻𝑒}_n(y).$

This expansion is most easily derived by using the two-dimensional Fourier transform of Template:Math, which is

$c(i{u}_1,i{u}_2)=\exp (-(u_1^2+u_2^2-2\rho u_1{u}_2)/2).$

This may be expanded as

$\exp (-(u_1^2+u_2^2)/2)\sum _{n=0}^{\mathrm{\infty }}\frac{{\rho }^n}{n!}(u_1{u}_2{)}^n.$

The Inverse Fourier transform then immediately yields the above expansion formula.

This result can be extended to the multidimensional case.

Fractional Fourier transform

Template:Main Since Hermite functions Template:Math are orthonormal eigenfunctions of the Fourier transform,

$ℱ[{\psi }_n](y)=(-i{)}^n{\psi }_n(y),$

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

$ℱ[f](y)=\int dxf(x)\sum _{n\ge 0}(-i{)}^n{\psi }_n(x){\psi }_n(y).$

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

${ℱ}_{\alpha }=\sum _{n\ge 0}(-i{)}^{2\alpha n/\pi }{\psi }_n(x){\psi }_n(y).$

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

The Mehler formula, for Template:Mvar = exp(−iTemplate:Mvar), thus directly provides

${ℱ}_{\alpha }[f](y)=\sqrt{\frac{1-i\cot (\alpha )}{2\pi }}{e}^{i\frac{\cot (\alpha )}{2}{y}^2}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{-i(\csc (\alpha )yx-\frac{\cot (\alpha )}{2}{x}^2)}f(x)\mathrm{d}x.$

The square root is defined such that the argument of the result lies in the interval [−π /2, π /2].

If Template:Mvar is an integer multiple of Template:Mvar, then the above cotangent and cosecant functions diverge. In the limit, the kernel goes to a Dirac delta function in the integrand, Template:Mvar or Template:Mvar, for Template:Mvar an even or odd multiple of Template:Mvar, respectively. Since ${ℱ}^2$[[[:Template:Mvar]] ] = Template:Mvar(−Template:Mvar), ${ℱ}_{\alpha }$[[[:Template:Mvar]] ] must be simply Template:Math or Template:Math for Template:Mvar an even or odd multiple of Template:Mvar, respectively.

See also

• Template:Slink
• Heat kernel
• Hermite polynomials
• Parabolic cylinder functions
• Template:Slink

References

• Nicole Berline, Ezra Getzler, and Michèle Vergne (2013). Heat Kernels and Dirac Operators, (Springer: Grundlehren Text Editions) Paperback Template:ISBN
• Template:Cite journal
• Template:Cite journal