CIM Seminar with Maurice de Gosson

Abstract
The physical question of "quantization" corresponds to the choice of a suitable pseudo-differential calculus satisfying certain properties. Born, Jordan (and Heisenberg) proposed in 1925 a rigorous method for quantizing monomials, which they extended to arbitrary functions. Their quantization scheme was, however, soon superseded by that of Weyl (the "Weyl correspondence"), which is easier to implement and has the property of symplectic covariance. The differences between Weyl and Born-Jordan quantization are of a rather subtle nature; in particular the Born-Jordan quantization is not injective, as opposed to the case of the Weyl correspondence.  We will discuss these differences, by redefining both quantization schemes in terms of Shubin's theory of parameter-dependent pseudo-differential operators. In our study the notion of "Cohen class" also plays a central role, and allows us to give an alternative definition of Born--Jordan operators using a modified version of the Wigner transform. We will also give precise asymptotic expansions of symbols belonging to Shubin's global symbol classes $\Gamma_{\rho}^{m}$, allowing to pass from Born-Jordan quantization to Weyl quantization, and vice-versa. Time permitting, we also briefly discuss some applications to time-frequency analysis (regularity results for Born--Jordan operators in Feichtinger's modulation spaces).

Registration
Register to the event here. The registration closes February 13.