Quantum ergodicity for Eisenstein functions (Q309760)

In the earlier work of the second author [J. Funct. Anal. 97, No. 1, 1--49 (1991; Zbl 0743.58034)], a quantum ergodicity theorem for Eisenstein series on a finite area hyperbolic surface with cusps is obtained. In this note, a simpler proof of that result is given. The new proof is at the same time more general, as it considers Eisenstein functions on an arbitrary cusp manifold with ergodic geodesic flow.NEWLINENEWLINEFor a semi-classical pseudo-differential operator \(\mathrm{Op}_h(\sigma)\) with compactly supported symbols \(\sigma\), the local Weyl law of \textit{W. Müller} [Math. Nachr. 125, 243--257 (1986; Zbl 0593.58042)], combined with the work of the first author [Commun. Math. Phys. 343, No. 1, 311--359 (2016; Zbl 1354.58031)], provides the average value of the matrix elements of the operator over the spectrum of the Laplace-Beltrami operator on the manifold. This note proves that its mean absolute deviation goes to zero, as \(h\) tends to zero.