Quantum Ergodicity

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 {(M,g)} is a compact Riemannian manifold with ergodic geodesic flow. Let {\{\phi_k\}} be the appropriately normalized eigenfunctions of the Laplacian {\Delta}. Then the measures with density {\frac 1 N\sum_{k=1}^N |\phi_k|^2 \cdot d{\mathrm{Vol}}} tend weakly to the usual volume {d{\mathrm{Vol}}}.

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 {d}-dimensional Riemannian manifold {M} with metric {g} (which will often be omitted from the notation).
  • (ii) The cotangent bundle {T^{\vee} M} which is naturally a symplectic manifold; the Poisson bracket for functions {f,g} on {T^{\vee} M} will be denoted {\{f,g\}}.
  • (iii) The Hilbert space {L^2(M)} of functions on {M} using the underlying volume form induced by the metric, and the space of linear operators {{\mathcal L}(L^2(M))} on it.
  • (iv) The spaces {C^{\infty}(T^{\vee} M)} and {C^\infty_0(T^{\vee} M)} of smooth, and smooth and compactly supported, functions on {T^{\vee} M}.
  • (v) Planck’s constant {\hbar} which will usually tend to zero and will be set to another parameter {E^{-1/2}}, where {E} 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 {\hbar {\rightarrow} 0}.

Theorem 1 There exists (a non-canonical) quantization map

\displaystyle  Op_\hbar : C^\infty_0(T^{\vee} M) {\rightarrow} {\mathcal L}(L^2(M)) \ \ \ \ \ (1)

satisfying the following properties:

  • (i) For composition of operators we have

    \displaystyle  Op_\hbar(a\cdot b) = Op_\hbar(a) \circ Op_\hbar(b) + O(\hbar) \ \ \ \ \ (2)

  • (ii) For commutators of operators we have

    \displaystyle  \left[ Op_\hbar(a) , Op_\hbar(b) \right] = i\hbar Op_\hbar(\{a,b\}) + O(\hbar^2) \ \ \ \ \ (3)

  • (iii) For Hilbert-Schmidt norms of operators we have

    \displaystyle  \| Op_\hbar(a) \|^2_{HS} \sim_{\hbar {\rightarrow} 0} \left(\frac{2\pi}{\hbar}\right)^{d} \|a\|^2_{L^2(T^{\vee} M)} \ \ \ \ \ (4)

  • (iv) The relationship to the Laplacian {\Delta} is as follows. Let {\chi:{\mathbb R}_{>0} {\rightarrow} {\mathbb R}} be a compactly supported function, and by abuse of notation set

    \displaystyle  \begin{array}{rcl}  \chi_M:T^{\vee} M {\rightarrow} {\mathbb R} \\ \xi \mapsto \chi(\|\xi\|) \end{array}

    Then we have

    \displaystyle  Op_\hbar(\chi_M) = \chi(-\hbar^2 \Delta) + O(\hbar) \ \ \ \ \ (5)

    where {\chi(-\hbar^2 \Delta)} is defined using operator calculus, using its action on eigenfunctions:

    \displaystyle  \chi(-\hbar^2 \Delta) \phi_k = \chi(-\hbar \lambda_k) \phi_k \ \ \ \ \ (6)

    assuming {\Delta \phi_k = \lambda_k \phi_k}.

A standard consequence of the existence of quantization is Weyl’s asymptotic law for the number of eigenvalues of the Laplacian. Namely, choose {\chi} to be identically {1} on the interval {[1,2]} and supported on {[1-\epsilon,2+\epsilon]}. Similarly, fix {\hbar = E^{-1/2}} where {d} is the dimension of {M}.

Then the quantization of {\chi_M} satisfies {Op_\hbar(\chi_M) = \chi(-E^{-d} \Delta) + O(\hbar)}. We can take traces on both sides (and ignoring terms of order {O(\hbar)} and {\epsilon})

\displaystyle  \begin{array}{rcl}  tr \left( Op_\hbar(\chi_M)^2 \right)\sim & tr \left(\chi(E^{-d} \Delta)^2\right)\\ \sim & \sum_{E\leq \lambda_k \leq 2 E} \chi\left(\frac{\lambda_k}{E}\right)\\ \sim & N(E) \end{array}

where {N(E)} is the number of eigenvalues of {\Delta} in {[E,2E]}. Now for a self-adjoint operator, the trace of its square is the same as the Hilbert-Schmid norm, so we have

\displaystyle  \begin{array}{rcl}  {\mathrm{tr}} \left( Op_\hbar(\chi_M)^2 \right) = & \| Op_\hbar(\chi_M) \|^2_{HS}\\ \sim & (2\pi)^d \cdot \hbar^{-d} \int_{T^{\vee} M} \chi_M^2 d{\mathrm{Vol}}\\ \sim & (2\pi)^d \cdot E^d \cdot {\mathrm{Vol}}(M) \end{array}

We therefore obtain Weyl’s law:

\displaystyle  N(E) \sim E^d \cdot (2\pi)^d {\mathrm{Vol}}(M) \ \ \ \ \ (7)

Remark 2 The quantization map can be defined on a larger subset than just smooth compactly supported functions on {T^{\vee} M}. 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 {{\mathcal H}:T^{\vee} M {\rightarrow} {\mathbb R}} and its quantization {H:=Op_\hbar({\mathcal H})}.

Classical dynamics is concerned with the flow {g_t} defined by {{\mathcal H}} using the vector field {d{\mathcal H}^\#}. Here {d{\mathcal H}} is the differential of {{\mathcal H}} and {(-)^\#} denotes the isomorphism induced by the symplectic form between {1}-forms and vector fields. The evolution of functions is described using the Poisson bracket:

\displaystyle  \dot{f}:= \{{\mathcal H},f\} \textrm{ for }f\in C^\infty(T^{\vee} M) \ \ \ \ \ (8)

Quantum dynamics occurs on {L^2(M)}. A function {\phi \in L^2(M)} evolves according to the Schrödinger equation

\displaystyle  \dot{\phi}:= \frac{-i}{\hbar}H\cdot \phi \ \ \ \ \ (9)

Similarly, operators {A\in {\mathcal L}(L^2(M))} evolve using commutators:

\displaystyle  \dot{A}:= \frac{-i}{\hbar}[H,A] \ \ \ \ \ (10)

Finally, we denote the time evolution operator by

\displaystyle  U(t):= \exp\left(\frac{-i}{\hbar} t H\right) \ \ \ \ \ (11)

The next result, known as Egorov’s theorem, allows one to compare the classical and quantum evolutions.

Theorem 3 Consider a function {a\in C^\infty(M)}. Denote by {a_t:=g_{-t}^* a} its time evolution under the classical flow. Consider also its quantization {A:=Op_\hbar(a)} and its quantum evolution {A_t := U(t) A U(-t)}. Then we have

  • Infinitesimal Comparison:

    \displaystyle  \frac {d}{dt}\mid_{t=0} A_t = Op_\hbar\left(\frac {d}{dt}\vert_{t=0} a_t \right) + O(\hbar) \ \ \ \ \ (12)

  • Finite Time Comparison:

    \displaystyle  A_t = Op_\hbar(a_t) + O(t\cdot \hbar) \ \ \ \ \ (13)

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

\displaystyle  \begin{array}{rcl}  \frac{d}{dt}\vert_{t=0} A_t = & \frac{-i}{\hbar}[H,A]\\ = & \frac{-i}{\hbar}\left( i\hbar Op_\hbar(\{{\mathcal H},a\}) + O(\hbar^2) \right)\\ = & Op_\hbar(\{{\mathcal H},a\}) + O(\hbar) \end{array}


— 3. Quantum Ergodicity —

We keep the same notation as in previous sections. In particular, {N(E)} denotes the number of eigenvalues of {\Delta} in the interval {[E,2E]}.

Theorem 4 Let {\{\phi_k\}} denote the collection of normalized eigenfunctions of {\Delta} with eigenvalues {\{\lambda_k\}}. Each {\phi_k} gives a measure {|\phi_k|^2 d{\mathrm{Vol}}} on {M}. Assume the geodesic flow is ergodic on {M}. We then have the following weak convergence of measures

\displaystyle  \frac{1}{N(E)}\left(\sum_{\lambda_k \in [E,2E]} |\phi_k|^2 \right) d{\mathrm{Vol}} {\rightarrow} d{\mathrm{Vol}} \ \ \ \ \ (14)

Before proceeding to the proof, we need to make a comment about quantization of functions which are not necessarily compactly supported. Namely, let {a\in C^\infty(M)} be a smooth function {M}. By pull-back to {T^{\vee} M} it gives a smooth function on the cotangent space, constant in each fiber. Its quantization {Op_\hbar(a)} is well-defined and agrees with the operator of pointwise multiplication by the function {a}. 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 {a} under the flow will be denoted

\displaystyle  a_t(x,\xi) := a(g_t(x,\xi)) = g_{-t}^*a(x,\xi) \ \ \ \ \ (15)

Note that this is no longer a function which is constant on the fibers of {T^{\vee} M{\rightarrow} M}. Similarly, denote the time averages of {a} by:

\displaystyle  Avg_T (a) := \frac 1 T \left(\int_{0}^T a_t dt\right) \in C^\infty(T^{\vee} M) \ \ \ \ \ (16)

Proof: Because the eigenfunctions are normalized, to check the claim it suffices to prove that for any {a\in C^\infty(M)} such that {\int_M a\, d{\mathrm{Vol}} =0} we have

\displaystyle  \frac 1 {N(E)} \sum_{\lambda_k \in [E,2E]} | \langle a \phi_k , \phi_k\rangle |^2 {\rightarrow} 0 \ \ \ \ \ (17)

Here {a\phi_k} denotes the multiplication of {\phi_k} and {a} as functions on {M}. By the ergodicity of the geodesic flow and our assumption on {a}, the time-averages of {a} satisfy:

\displaystyle  Avg_T(a) {\rightarrow} 0 \textrm{ pointwise a.e.} \ \ \ \ \ (18)

From the earlier discussion on the quantization of {a} we have

\displaystyle  \langle a\phi_k , \phi_k \rangle = \langle Op_\hbar(a) \phi_k, \phi_k \rangle \ \ \ \ \ (19)

Now, using that {U(t)} is a self-adjoint operator we have

\displaystyle  \begin{array}{rcl}  \langle Op_\hbar(a) \phi_k, \phi_k \rangle = & \langle U(t) Op_\hbar(a) U(-t) \phi_k, \phi_k \rangle\\ = & \langle Op_\hbar(a_t) \phi_k, \phi_k \rangle + O(t\cdot \hbar) \end{array}

where to transition to the last line we used Egorov’s theorem. We can now average the above expression over time to find:

\displaystyle  \langle Op_\hbar(a) \phi_k, \phi_k \rangle = \langle Op_\hbar(Avg_T(a)) \phi_k, \phi_k \rangle + O(T\cdot \hbar) \ \ \ \ \ (20)

Note that {Avg_T(a)} 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 {\chi} supported in {[1-\epsilon, 2 + \epsilon]}. Then we have

\displaystyle  \chi(-\hbar^2 \Delta) \phi_k = \chi(-\hbar^2 \lambda_k) \phi_k \ \ \ \ \ (21)

We can then write

\displaystyle  \begin{array}{rcl}  \chi(-\hbar^2 \lambda_k) \langle a\phi_k , \phi_k \rangle = & \chi(-\hbar^2 \lambda_k) \langle Op_\hbar(Avg_T(a)) \phi_k, \phi_k \rangle + O(T\cdot \hbar)\\ = & \langle Op_\hbar(Avg_T(a)) \chi(-\hbar^2 \lambda_k) \phi_k, \phi_k \rangle + O(T\cdot \hbar)\\ = & \langle Op_\hbar(Avg_T(a)) Op_\hbar(\chi_M)) \phi_k, \phi_k \rangle + O(T\cdot \hbar)\\ = & \langle Op_\hbar(Avg_T(a) \cdot \chi_M)) \phi_k, \phi_k \rangle + O(T\cdot \hbar)\\ \end{array}

Recall that by the rules of quantization, we have for the Hilbert-Schmidt norm:

\displaystyle  \| Op_\hbar(Avg_T(a) \cdot \chi_M)) \|^2_{HS} \sim \hbar^{-d} \| Avg_T \cdot \chi_M\|^2_{L^2(T^{\vee} M)} \ \ \ \ \ (22)

Finally, recall that to compute the Hilbert-Schmidt norm of a self-adjoint operator {T}, it suffices to pick an orthonormal basis {\{e_k\}}:

\displaystyle  \| T \|^2_{HS} = \sum \langle T^2 e_k ,e_k \rangle = \sum \|T e_k\|^2 \ \ \ \ \ (23)

We can now combine our calculations to find (recall that {\hbar := E^{-1/2}}:

\displaystyle  \begin{array}{rcl}  \frac 1 {N(E)} \sum_{\lambda_k \in [E,2E]} |\langle a \phi_k, \phi_k|^2 \leq & \\ \leq & \frac 1{N(E)} \sum_{\lambda_k \in [E,2E]} |\langle a \cdot \chi(-\hbar^2 \lambda_k) \phi_k, \phi_k|^2 \\ \leq & \frac 1{N(E)} \sum_{\lambda_k \in [E,2E]}\left[ |\langle Op_\hbar(Avg_T(a)\cdot \chi )\phi_k, \phi_k|^2 + O(T\cdot \hbar) \right]\\ \end{array}

Finally, recall that {N(E)\sim E^{d/2}} so the last term above equals:

\displaystyle  \begin{array}{rcl}  E^{-d/2} \sum_{\lambda_k \in [E,2E]}\left[ |\langle Op_\hbar(Avg_T(a)\cdot \chi )\phi_k, \phi_k|^2 \right] + O(T\cdot \hbar)\\ \leq E^{-d/2} \| Op_\hbar(Avg_T(a)\cdot \chi) \|^2_{HS} + O(T\cdot \hbar)\\ \sim E^{-d/2} \hbar^d \| Avg_T(a)\cdot \chi \|_{L^2(T^{\vee} M)} + O(T\cdot \hbar)\\ = \| Avg_T(a)\cdot \chi \|_{L^2(T^{\vee} M)} + O(T\cdot \hbar) \end{array}

Now fix {T} such that {Avg_T(a)} is arbitrarily small in {L^2}-norm. Recall that {\hbar = E^{-1/2}}, from which the convergence to zero of the above term is clear as {\hbar {\rightarrow} 0} (or, equivalently, {E{\rightarrow} \infty}). \Box


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