Semiclassical limits for the hyperbolic plane
The author studies the concentration properties of high-energy eigenfunctions of the Laplacian on the upper half-plane with special emphasis on automorphic eigenfunctions (Maaß forms). Let \(\Gamma\) be a cofinite but noncompact subgroup of \(\text{SL}_2 (\mathbb{R})\) and \(\varphi\) a cuspidal eigenfunction of the Laplacian on \(\Gamma \setminus \mathbb{H}\). Then one important theme of the investigation is that at high energy the concentration measure on the space of geodesics can be approximately given in terms of certain coefficient sums of \(\varphi\) involving the Fourier coefficients of \(\varphi\). A first application to a congruence subgroup \(\Gamma\) yields that a certain spectral average of the aforementioned coefficient sums is asymptotically related with the area of \(\Gamma \setminus \mathbb{H}\). A second application concerns a reinterpretation of results of Luo and Sarnak and of Jakobson on quantum ergodicity of eigenfunctions, in particular of the modular Eisenstein series. In a third application a lower bound for certain square coefficient sums is given. At the center of investigation is the microlocal lift of an eigenfunction as introduced by S. Zelditch. Generalizing a result of Zelditch the author proves that the microlocal lift satisfies a partial differential equation; this equation is basic to the author's approach. Special attention is devoted to the microlocal lift of the modified Bessel function. At high energy, the microlocal lift of a modified Bessel function turns out to be concentrated along a single geodesic. Since the results of this long work cannot be given adequately in a few lines, the reader is advised to look at the author's careful and detailed introduction.
scientific article; zbMATH DE number 1820827
scientific article