Probability Seminar

Alexander MollUniversity of Massachusetts, Boston
Exact Results for the Quantum Benjamin-Ono Equation on the Torus

Monday, May 9, 2022 - 3:45pm
Malott 406

Abstract: In this talk, we use a fractional Gaussian field on the 1-dimensional real torus to quantize Benjamin-Ono waves and give a dynamical interpretation of a result of Stanley (1989). Precisely, the classical Benjamin-Ono equation on the torus is a non-linear and non-local model of dispersive waves. Starting from the standard Gaussian in the symplectic space of this equation, we construct the quantum Benjamin-Ono equation on the torus without any path integrals. Remarkably, a result of Stanley (1989) for Jack polynomials implies an exact description of the spectrum of the quantum Hamiltonian in this model. We prove that if one considers Bohr-Sommerfeld quantization of the classical Benjamin-Ono multi-phase solutions on the torus found by Satsuma-Ishimori (1979), then the resulting approximation of the quantum spectrum found by Stanley is exact after an explicit renormalization of the coefficient of dispersion predicted by Abanov-Wiegmann (2005).