Applications of a Commutation Formula II

The principal {\it commutation formula} underlying the paper by Deift [24], is of the type $$-z(AB-zI{)}^{-1}+A(BA-zI{)}^{-1}B=I,z\in [\rho (AB)\cap \rho (BA)]\mathrm{\backslash }\{0\},$$ where, temporarily, and for simplicity only, $A,B$ represent bounded operators in a Banach space. A fundamental consequence of this commutation formula is the fact that $AB$ and $BA$ are {\it essentially isospecral} in the sense that $$\sigma (AB)\mathrm{\backslash }\{0\}=\sigma (BA)\mathrm{\backslash }\{0\}\mathrm{.}$$ Moreover, under appropriate trace class hypotheses, this implies the fundamental identity for (Fredholm) determinants $$\det (I-AB)=\det (I-BA),$$ which extends to modified (or regularized) Fredholm determinants. In this paper we briefly recall some of the applications of these commutation formulas to isospectrality, integrable systems, regularization of PDEs, and estimates in Statistical Mechanics and Quantum Field Theory discussed in [24]. Subsequently, we review some of the ramifications of commutation formulas that were established over the past fifty years, including determinants and dimension reduction, random matrix theory and dimension expansion, integrable operators, Poncelet’s porism, interpolation, eigenvalue computation, KPZ, Pfaffians, the focusing Gross-Pitaevskii equation, one-dimensional scattering theory, Floquet theory, and the Miura transformation connecting the KdV and mKdV hierarchies.