BZ Seminar in Analysis: Aleksandr Logunov (University of Geneva)

Sign of eigenfunctions

The functions sin(kx), cos(kx) are positive on half of the circle and are negative on another half. D.Armitage and S.Gardiner conjectured that the sign of spherical harmonics is always positive on a portion of the sphere bounded below by a positive constant, which depends only on the dimension of the sphere. This phenomenon is called quasi-symmetry of sign and it was proved by H.Donnelly and C.Fefferman. Nazarov, Polterovich and Sodin suggested that quasi-symmetry of sign happens on small scales in the regime when the eigenvalue grows to infinity. We will talk about the distribution of sign based on a joint work in progress with Fedya Nazarov.

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