Quantum unique ergodicity for \(\text{SL}_2(\mathbb Z)\setminus \mathbb H\). (English) Zbl 1209.58019

The author studies some aspects of the so-called quantum unique ergodicity. Let us recall that, if \(M\) is a compact finite dimensional Riemannian manifold and \(\{\psi_n\}_{1\leq n\leq \infty}\) is an orthonormal basis in \(L^2(M)\), consisting of eigenfunctions of the Laplacian \(\triangle\) of \(M\), to each \(\psi_n\), a measure \(\bar\mu(\psi_n)=|\psi_n|^2\,d\rho\) can be associated, or a distribution, on \(M\). Here, \(d\rho\) is the canonical measure on the Riemannian manifold \(M\). Let \(S^*M\subset T^*M\) be the unit cotangent bundle on \(M\). To each \(\psi_n\), one can associate a distribution \(\mu(\psi_n)\) on \(S^*M\) projecting to \(\bar\mu(\psi_n)\) (microlocal lift) (A. I. Shnirel’man (1974)). A measure \(\mu\) on \(S^*M\) is called a (microlocal) quantum limit if it is a weak-limit of a sequence of distributions \(\mu_n\), microlocal lift of a sequence of eigenfunctions \(\psi_n\) with \(|\lambda_n|\to \infty\), where \(\lambda_n\) is an eigenvalue for the eigenfunction \(\psi_n\).
The following theorem characterizes the quantum limit (Shnirel’man-Zelditch (1986), Colin de Verdière (1985)). Let the eigenfunctions \(\{\psi_n\}_{1\leq n\leq \infty}\) be ordered by increasing eigenvalues. Then: 7mm
\(\text{weak-lim}_{N\to\infty}{{1}\over{N}}\sum_{1\leq n\leq N}\mu_n=d\lambda\), where \(d\lambda\) is the Liouville measure on \(S^*M\);
if the geodesic flow on \(S^*M\) is ergodic, there exists a subsequence \(\{\psi_k\}_{1\leq k\leq \infty}\) of density 1, such that \(\text{weak-lim}_{k\to\infty}\mu_k=d\lambda\). For this subsequence, \(\text{weak-lim}_{k\to\infty}\bar\mu_k=d\rho\).

Since it is well-known that the geodesic flow on a manifold of (constant) negative curvature is ergodic (E. Hopf (1939), D. V. Anosov (1967)), the following conjecture has been stated.
(Rudnick-Sarnak’s quantum unique ergodicity conjecture (1994)) Let \(M\) be a compact manifold of strictly negative sectional curvature. Then \(\bar\mu_n\) weakly converges to the measure \(d\rho\) and \(d\lambda\) is the unique quantum limit on \(S^*M\).
If \(M\) is a hyperbolic surface of constant negative curvature, the universal cover \(\widetilde{M}\) of \(M\) is the upper half-plane: \(\widetilde{M}\cong {\mathbb H}\equiv\{z=x+iy\in\mathbb C\mid y>0\}\subset\mathbb C\). The group \(\text{SL}_2(\mathbb R)\) acts on \(\mathbb H\) by Möbius transformations. One has \(\mathbb H\backsimeq \text{PSL}_2(\mathbb R)/\text{SO}_2(\mathbb R)\), and the isomorphism \(S^*M\cong X\), where \(X\equiv \Gamma\backslash\text{PSL}_2(\mathbb R) \to M=\Gamma\backslash{\mathbb H}\) is the \(\text{SO}_2(\mathbb R)\simeq S^1\) bundle, for a uniform lattice (congruence lattice) \(\Gamma< \text{PSL}_2(\mathbb R)\). Then the geodesic flow on \(S^*M\) is parametrized by the action of the maximal split torus on \(X\). Furthermore, there are additional operators (Hecke operators) \(T_p: L^2(X)\to L^2(X)\), for each prime \(p\) (except for a finite set of ramified primes) commuting with the right action of \(\text{PSL}_2(\mathbb R)\). \(T_p\) acts also on functions on \(M\) and commutes with \(\triangle\). The Hecke-Maass forms are just the joint eigenfunctions of all the Hecke operators \(T_p\) and \(\triangle\). An arithmetic quantum limit is a quantum limit \(\mu_\infty\) obtained from micro-local lifts of Hecke-Maass forms. The arithmetic quantum chaos problem is the classification of such limits. One has the following theorem (Lindenstrauss-theorem (2003)). Let \(M=\Gamma\backslash{\mathbb H}\) be a congruence quotient, and \(\mu_\infty\) an arithmetic quantum limit on \(X=\Gamma\backslash \text{PSL}_2(\mathbb R)\simeq S^*M\). Then \(\mu_\infty=c\,dx\), for some \(c\in[0,1]\), where \(dx\) is the Haar measure. If \(\Gamma\) is co-compact (i.e., arising from a quaternionic algebra), then \(c=1\).
In particular, the author considers quantum unique ergodicity for the modular surface \(M\equiv\text{SL}_2(\mathbb Z)\backslash{\mathbb H}\). The main result is a proof of Rudnick-Sarnak’s quantum unique ergodicity for normalized Hecke-Maass forms \(\phi\) on \(M\) (hence with unitary Peterson norm: \(\int_M|\phi(z)|^2{{1}\over{y^2}}\,dx\, dy=1\)). In other words, for every Maass form \(\phi\) the corresponding measure \(\mu(\phi)\) approaches the uniform distribution measure \({{3}\over{\pi}}{{1}\over{y^2}}\,dx\, dy\) as \(\lambda\to\infty\). (This generalizes a result by S. Zelditch (1991) for a typical Maass form \(\phi\), that is, the \(L^2\)-mass is uniformly spread out over a fundamental domain of \(\text{SL}_2(\mathbb Z)\backslash{\mathbb H}\).)
Lindenstrauss’ results prove that the only possible weak limits of the measure \(\mu(\phi)\) are the forms \(c\, \left(\frac 3{\pi}\right)\left(\frac 1{y^2}\right)\,dx\,dy\), where \(c\) is some constant in \([0,1]\). Therefore, Lindenstrauss established quantum unique ergodicity for \(M\) except for the possibility that, for some infinite sequence of Hecke-Maass forms \(\phi\), some of the \(L^2\) mass of \(\phi\) could “escape” into the cusp of \(M\). In this paper, the author eliminates the possibility of the escape of mass, and, therefore, he is able to complete the proof of quantum unique ergodicity for \(M\).
The paper, after a detailed introduction (where are summarized the main result, the principal steps for its proof and enough references to where the reader can find more accounts of the quantum unique ergodicity problem), splits into three more sections. 2. Deducing Theorem 1 and Proposition 2 from Theorem 3. (Theorem 3 and Proposition 2 are intermediate results used to obtain the main result contained in Theorem 1.) 3. Preliminaries for the proof of Theorem 3. Proof of Theorem 3.
Remark: Let us emphasize that the Rudnick-Sarnak conjecture proven in this paper for the modular surface \(M\equiv \text{SL}_2(\mathbb Z)\backslash{\mathbb H}\) is of great interest in number theory. In fact, the study of functions on \(\mathbb H\) that are invariant (or have a suitable transformation) under the action of \(\text{SL}_2(\mathbb Z)\) is related to modular forms. The latter are related to the well-known Taniyama-Shimura conjecture (1955), proved by A. Wiles (1995), since it states that all rational elliptic curves are also modular ones. The proof of this conjecture allowed A. Wiles to prove also Fermat’s last theorem about the non-existence of integer solutions of the Diophantine equation \(x^n+y^n=z^n\), \(n>2\), \(x,y,z\neq 0\). (Important subjects, related to the theory of cusp modular forms, and, hence to this paper, are the Ramanujan’s conjecture (1920) and Weil’s conjecture (1949), proven by P. Deligne (1973–74).)


11F72 Spectral theory; trace formulas (e.g., that of Selberg)
58J51 Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity
11F25 Hecke-Petersson operators, differential operators (one variable)
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