Young's geodesic QUE conjecture for restricted eigenfunctions

From papers

Let uju_j be even Hecke–Maass forms normalized by the paper's L2L^2 normalization, and let ff be an L2L^2-normalized holomorphic Hecke cusp form of weight kk. If ψ:R+R\psi:\mathbb{R}^+\to\mathbb{R} is smooth and compactly supported, Young's geodesic QUE conjecture.

limj0uj(iy)2ψ(y)dyy=02ψ(y)dyy,\lim_{j\to\infty}\int_0^\infty|u_j(iy)|^2\psi(y)\frac{dy}{y}=\int_0^\infty2\psi(y)\frac{dy}{y},

and

limk0ykf(iy)2ψ(y)dyy=0ψ(y)dyy.\lim_{k\to\infty}\int_0^\infty y^k|f(iy)|^2\psi(y)\frac{dy}{y}=\int_0^\infty\psi(y)\frac{dy}{y}.

The conjecture concerns equidistribution after restriction to the vertical geodesic joining 00 and ii\infty. The paper proves the analogous statement for Eisenstein series, whereas the corresponding cusp-form assertions are presented as conjectural.

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Sources & referencesView supporting material

Primary source

Matthew P. Young, “The quantum unique ergodicity conjecture for thin sets”, arXiv:1306.1554 (2013).

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