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Quantum ergodicity of Eisenstein series for arithmetic 3-manifolds. (English) Zbl 0982.11030

Let \(K\) be an imaginary quadratic number field with class-number one and denote by \({\mathfrak o}_K\) the ring of integers of \(K\). The group \(\text{PSL}(2,{\mathfrak o}_K)\) acts on the three-dimensional upper half-space \(\mathbb{H}^3\) and \(X:=\text{PSL}(2,{\mathfrak o}_K)\setminus\mathbb{H}^3\) has finite hyperbolic volume. Extending a quantum ergodicity result by W. Luo and P. Sarnak [Publ. Math., Inst. Hautes Étud. Sci. 81, 207-237 (1995; Zbl 0852.11024)] the author proves the following Theorem:
For all compact Jordan measurable subsets \(A,B\subset X\) we have \[ \lim_{t\to\infty}\frac{\mu_t(A)}{\mu_t(B)}=\frac{V(A)}{V(B)} , \] where \(V\) is the hyperbolic volume measure on \(\mathbb{H}^3\) and \(d\mu_t(P)=|E(P,1+it)|^2dV(P)\) with \(E(P,s)\) being the Eisenstein series for \(\text{PSL}(2,{\mathfrak o}_K)\). In fact, the author even proves that \[ \mu_t(A)\sim\frac{2V(A)}{\zeta_K(2)} \log t\quad\text{as}\quad t\to\infty. \]

MSC:

11F72 Spectral theory; trace formulas (e.g., that of Selberg)
58J51 Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)

Citations:

Zbl 0852.11024
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