Modern trends in Operator Theory and Geometric Analysis

Magnetic Laplacian and the conformal class of a surface

Not scheduled

1h

Aula Dini (Palazzo del Castelletto)

Speaker

Alessandro Savo (Sapienza Università di Roma)

Description

On a closed Riemannian surface we study the magnetic Laplacian with magnetic potential given by a harmonic 1-form A. Its lowest eigenvalue (magnetic ground state energy) is positive, unless A represents an integral cohomology class. We isolate a countable set of ground state energies which we call ''ground state spectrum'' of the metric. Our main result is to show that the ground state spectrum determines the volume and the conformal class of the metric. In particular, hyperbolic metrics are distinguished by their ground state spectrum (which is not true for the classical, non-magnetic Laplace spectrum).
Joint work with Bruno Colbois and Luigi Provenzano.