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Boundary-adapted arithmetic random waves

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Oleksiy Klurman
KTH Stockholm/MPIM
Die, 2019-11-12 14:00 - 15:00
MPIM Lecture Hall
In this talk, we test M. Berry's ansatz on nodal deficiency in presence of boundary. The square billiard is studied, where the high spectral degeneracies allow for the introduction of a Gaussian ensemble of random Laplace eigenfunctions ("boundary-adapted arithmetic random waves"). As a result of a precise asymptotic analysis, two terms in the asymptotic expansion of the expected nodal length are derived, in the high energy limit along a generic sequence of energy levels.
In particular, we shall focus on a number-theoretic aspect of this problem, describing the techniques introduced by E. Bombieri and J. Bourgain to study additive equations for integral points on the circles. This is based on a joint work with V. Cammarota and I. Wigman.


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