On a Gauss-Givental Representation of Quantum Toda Chain Wave Function

TL;DR

This paper provides a group-theoretic interpretation of Givental's integral representation of quantum Toda chain wave functions, connecting it with differential operators, recursive structures, and Baxter Q-operators.

Contribution

It constructs a new representation of the universal enveloping algebra in terms of differential operators, linking Givental's integral form with algebraic and recursive structures.

Findings

Established a differential operator representation of $U(rak{gl}(N))$

Connected Givental's integral with the Gauss decomposition

Discussed extensions to infinite and periodic Toda chains

Abstract

We propose group theory interpretation of the integral representation of the quantum open Toda chain wave function due to Givental. In particular we construct the representation of $U((\mathfrak{gl}(N))$ in terms of first order differential operators in Givental variables. The construction of this representation turns out to be closely connected with the integral representation based on the factorized Gauss decomposition. We also reveal the recursive structure of the Givental representation and provide the connection with the Baxter $Q$-operator formalism. Finally the generalization of the integral representation to the infinite and periodic quantum Toda wave functions is discussed.