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Maffucci, Riccardo W.

Restriction of 3D arithmetic Laplace eigenfunctions to a plane. (English) Zbl 1457.60073

Electron. J. Probab. 25, Paper No. 60, 17 p. (2020).
Summary: We consider a random Gaussian ensemble of Laplace eigenfunctions on the 3D torus, and investigate the 1-dimensional Hausdorff measure (‘length’) of nodal intersections against a smooth 2-dimensional toral sub-manifold (‘surface’). A prior result of ours prescribed the expected length, universally proportional to the area of the reference surface, times the wavenumber, independent of the geometry.
In this paper, for surfaces contained in a plane, we give an upper bound for the nodal intersection length variance, depending on the arithmetic properties of the plane. The bound is established via estimates on the number of lattice points in specific regions of the sphere.

MSC:

60G60 Random fields
11P21 Lattice points in specified regions
60G15 Gaussian processes
58J50 Spectral problems; spectral geometry; scattering theory on manifolds

Keywords:

nodal intersections; arithmetic random waves; lattice points on spheres; Gaussian random fields; Kac-Rice formulas
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\textit{R. W. Maffucci}, Electron. J. Probab. 25, Paper No. 60, 17 p. (2020; Zbl 1457.60073)
Full Text: DOI arXiv Euclid

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