The Quantum Ergodicity Theorem of Shnirelman, Colin de Verdière and Zelditch is a result that combines in a nice way several different concepts. Namely, suppose is a compact Riemannian manifold with ergodic geodesic flow. Let be the appropriately normalized eigenfunctions of the Laplacian . Then the measures with density tend weakly to the usual volume .
I find the result interesting because it makes a dynamical assumption on the metric and says something about the eigenfunctions of the Laplacian. The tool connecting these two aspects is microlocal analysis. It appears in the proof, but not in the statement of the result.
Below the fold is a brief discussion of the main ideas that go into the proof. Weyl’s law for the density of eigenvalues appears as a rather simple result of the general formalism of microlocal analysis.
I am following a series of lectures notes due to Faure and Anantharaman available here.
Fix the following notation for the rest of the post:
- (i) A -dimensional Riemannian manifold with metric (which will often be omitted from the notation).
- (ii) The cotangent bundle which is naturally a symplectic manifold; the Poisson bracket for functions on will be denoted .
- (iii) The Hilbert space of functions on using the underlying volume form induced by the metric, and the space of linear operators on it.
- (iv) The spaces and of smooth, and smooth and compactly supported, functions on .
- (v) Planck’s constant which will usually tend to zero and will be set to another parameter , where will be the energy cut-off.
— 1. Basic microlocal analysis —
Basic principles of quantization suggest that to a function on phase space (an observable) one should associated a hermitian self-adjoint operator on some space of functions. Under this assignment, the Poisson bracket of two functions should go to the commutator of the operators.
A starting point of microlocal analysis is a recipe for constructing such a passage from functions to operators. However, exact quantization is rarely possible and one has to work with statements that only hold in the limit .
Theorem 1 There exists (a non-canonical) quantization map
satisfying the following properties:
- (i) For composition of operators we have
- (ii) For commutators of operators we have
- (iii) For Hilbert-Schmidt norms of operators we have
- (iv) The relationship to the Laplacian is as follows. Let be a compactly supported function, and by abuse of notation set
Then we have
where is defined using operator calculus, using its action on eigenfunctions:
A standard consequence of the existence of quantization is Weyl’s asymptotic law for the number of eigenvalues of the Laplacian. Namely, choose to be identically on the interval and supported on . Similarly, fix where is the dimension of .
Then the quantization of satisfies . We can take traces on both sides (and ignoring terms of order and )
where is the number of eigenvalues of in . Now for a self-adjoint operator, the trace of its square is the same as the Hilbert-Schmid norm, so we have
We therefore obtain Weyl’s law:
Remark 2 The quantization map can be defined on a larger subset than just smooth compactly supported functions on . Under appropriate growth conditions at infinity, most smooth functions can be quantized. Such quantizations will appear below.
— 2. Hamiltonians, the Schrödinger equation, and Egorov’s theorem —
Fix a Hamiltonian and its quantization .
Classical dynamics is concerned with the flow defined by using the vector field . Here is the differential of and denotes the isomorphism induced by the symplectic form between -forms and vector fields. The evolution of functions is described using the Poisson bracket:
Quantum dynamics occurs on . A function evolves according to the Schrödinger equation
Similarly, operators evolve using commutators:
Finally, we denote the time evolution operator by
The next result, known as Egorov’s theorem, allows one to compare the classical and quantum evolutions.
Theorem 3 Consider a function . Denote by its time evolution under the classical flow. Consider also its quantization and its quantum evolution . Then we have
- Infinitesimal Comparison:
- Finite Time Comparison:
Note that the finite time comparison is simply an integrated version of the infinitesimal comparison.
Proof: (Sketch) For the infinitesimal version, one just needs to note that
— 3. Quantum Ergodicity —
We keep the same notation as in previous sections. In particular, denotes the number of eigenvalues of in the interval .
Theorem 4 Let denote the collection of normalized eigenfunctions of with eigenvalues . Each gives a measure on . Assume the geodesic flow is ergodic on . We then have the following weak convergence of measures
Before proceeding to the proof, we need to make a comment about quantization of functions which are not necessarily compactly supported. Namely, let be a smooth function . By pull-back to it gives a smooth function on the cotangent space, constant in each fiber. Its quantization is well-defined and agrees with the operator of pointwise multiplication by the function . Note that it is a bounded operator, but it does not have bounded Hilbert-Schmid norm!
Finally, we shall use the following notation. The evolution of under the flow will be denoted
Note that this is no longer a function which is constant on the fibers of . Similarly, denote the time averages of by:
Proof: Because the eigenfunctions are normalized, to check the claim it suffices to prove that for any such that we have
Here denotes the multiplication of and as functions on . By the ergodicity of the geodesic flow and our assumption on , the time-averages of satisfy:
From the earlier discussion on the quantization of we have
Now, using that is a self-adjoint operator we have
where to transition to the last line we used Egorov’s theorem. We can now average the above expression over time to find:
Note that goes to zero pointwise, but could in principle have large operator or Hilbert-Schmidt norm. To address this, we shall impose a cutoff in the fiber directions. As in the derivation of Weyl’s law, pick a bump function supported in . Then we have
We can then write
Recall that by the rules of quantization, we have for the Hilbert-Schmidt norm:
Finally, recall that to compute the Hilbert-Schmidt norm of a self-adjoint operator , it suffices to pick an orthonormal basis :
We can now combine our calculations to find (recall that :
Finally, recall that so the last term above equals:
Now fix such that is arbitrarily small in -norm. Recall that , from which the convergence to zero of the above term is clear as (or, equivalently, ).