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On the topography of Maass waveforms for \(\text{PSL}(2,\mathbb Z)\). (English) Zbl 0813.11035

The paper provides “a glimpse into arithmetic quantum chaos”. In particular, one finds some beautiful color contour maps for eigenfunctions \(\phi_ n\) of the Laplace operator on the fundamental domain of the modular group \(\text{SL}(2, \mathbb Z)\). Nodal lines \((\phi_ n = 0)\) are also plotted. Physicists have asked whether we can see ridges or scars in contour maps like these as the eigenvalue goes to infinity. And the question is whether the scarring is along geodesics. See, for example, M. C. Gutzwiller [Chaos in classical and quantum mechanics. New York etc.: Springer-Verlag (1990; Zbl 0727.70029)]. Recently Sarnak and others have addressed this question [see P. Sarnak, Arithmetic quantum chaos, The Schur lectures (1992). Ramat-Gan: Bar-Ilan University, Isr. Math. Conf. Proc. 8, 183–236 (1995; Zbl 0831.58045)]. In studying the pictures the authors find, for example, that there do seem to be ridges but they do not seem to lie along geodesics.
Other questions are also asked. Look at the \(n\)-th eigenfunction \(\phi_ n\) of \(\Delta\) on \(L^ 2 (\text{SL}(2, \mathbb Z) \backslash H)\) and set \[ \nu_{n,A} (E) = \mu (A)^{-1} \mu \left\{ z \in A \mid \phi_ n (z) \in E \right\}. \] Does this converge nicely to a Gaussian distribution? The authors find that this seems to be true by studying histograms of the value distribution of \(\phi_ n\) for large \(n\).

MSC:

11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11F11 Holomorphic modular forms of integral weight
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