Frozen Gaussian Approximation

The purpose:

I will show you how to carry out asymptotic analysis calculations for the semiclassical Schrodinger equation:

The benefit for using asymptotic analysis in computing will be clear in this example calculation, namely we will change the original problem of computing the solution to a PDE into the new problem of computing the solution to an ODE.

First, we begin by describing what is the Frozen Gaussian Approximation: Loosely speaking, it is a superposition of Gaussian functions of fixed width in phase space. If we have a complex valued L2 function, we are able to represent said function using a superposition of infinitely many Gaussian functions. That is the contents of our first theorem:

Constructing the FGA ansatz:

Now we want to extend this to functions of both x and t, since the time-dependent Schrodinger equation is obviously time-dependent. Thus, I will do the following modifications to equation (1).

-Change psi(x) on the left to psi(x,t)

-Change psi(y) on the right to the initial condition psi_{0}(y,0)

-Allow the Gaussians (the ones depending on x) to move locations in phase space i.e. Let q,p depend on time: (Q(t,q,p),P(t,q,p)).
-I will also specify the movement of (Q,P) via a Hamiltonian flow.

-Multiply by an amplitude term a(t,q,p). This term will also be a function of time.

-Multiply by the phase term e^(i/epsilon S) with S=S(t,q,p).

Why do we expect these modifications to work? That is a story for another day, but it is a natural extension of the WKB method and Gaussian beam method.

Crucial Lemma:

We need one more thing before we carry out the asymptotic analysis calculations. We will Taylor expand the potential in the Schrodinger equation about Q. Thus, there will be terms (x-Q) raised to arbitrary powers. These terms will then be of order epsilon to some power (to be described below). Of course, we need to know what order of epsilon each term is before we start collecting orders of epsilon.

As we will see later on, the S equation is derived by collecting the O(1) terms and, the a equation will be derived from the O(epsilon) terms.

And now on to the Lemma and its proof:

Main Derivation:

Now we are ready to do asymptotic analysis. Essentially, we will plug in the FGA ansatz into the Schrodinger equation and collect orders of epsilon.

Next we Taylor expand and use the lemma:

Collect O(1) terms and O(epsilon) terms. The terms in the integrand will have to vanish. I wont work out the O(epsilon^2) terms but, the calculation is essentially the same. Taylor expand to higher order to get better corrections and apply the Lemma:

And that concludes the lowest order approximation for using the FGA for the semiclassical Schrodinger equation. The FGA can be applied to other equations as well, the main difference is that the Hamiltonian might be different. However, a similar calculation can be carried out for these other equations i.e. use Taylor expansion and apply the Lemma.

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