Spectral theory for the $q$-Boson particle system. (17th September 2014)

• Record Type: Journal Article
• Authors: Borodin, Alexei; Corwin, Ivan; Petrov, Leonid; Sasamoto, Tomohiro
• Is Part Of: Compositio mathematica. Volume 151:Number 1(2015:Jan.)
• Page Start: 1
• Page End: 67
• Publication Date: 2014-09-17
• Subjects: 05E05, -- 82B23 (primary); Bethe ansatz, -- Plancherel theory, -- quantum integrable systems, -- q-Boson; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X14007532
• Languages: English
• ISSNs: 0010-437X

Abstract

Abstract: We develop spectral theory for the generator of the $q$ -Boson (stochastic) particle system. Our central result is a Plancherel type isomorphism theorem for this system. This theorem has various implications. It proves the completeness of the Bethe ansatz for the $q$ -Boson generator and consequently enables us to solve the Kolmogorov forward and backward equations for general initial data. Owing to a Markov duality with $q$ -TASEP ( $q$ -deformed totally asymmetric simple exclusion process), this leads to moment formulas which characterize the fixed time distribution of $q$ -TASEP started from general initial conditions. The theorem also implies the biorthogonality of the left and right eigenfunctions. We consider limits of our $q$ -Boson results to a discrete delta Bose gas considered previously by van Diejen, as well as to another discrete delta Bose gas that describes the evolution of moments of the semi-discrete stochastic heat equation (or equivalently, the O'Connell–Yor semi-discrete directed polymer partition function). A further limit takes us to the delta Bose gas which arises in studying moments of the stochastic heat equation/Kardar–Parisi–Zhang equation.