Liouville Quantum Mechanics on a Lattice Large from Geometry of Quantum Lorentz Group

TL;DR

This paper explores the quantum Lobachevsky space and its relation to Liouville quantum mechanics, deriving a difference operator on a lattice that converges to a Schrödinger Hamiltonian, with solutions linked to special polynomials.

Contribution

It introduces a quantum analog of horospheric coordinates on ${f L}_q^3$ and connects the spectrum of a difference operator to $q$-Hermite polynomials, bridging quantum groups and Liouville theory.

Findings

Derived a second order difference operator on a lattice from quantum group actions.

Identified eigenfunctions as $q$-Hermite polynomials related to Macdonald polynomials.

Connected the spectrum to scattering in the $Z_N$ Baxter model in a special limit.

Abstract

We consider the quantum Lobachevsky space ${\bf L}_q^3$, which is defined as subalgebra of the Hopf algebra ${\cal A}_q(SL_2({\bf C}))$. The Iwasawa decomposition of ${\cal A}_q(SL_2({\bf C}))$ introduced by Podles and Woronowicz allows to consider the quantum analog of the horospheric coordinates on ${\bf L}_q^3$. The action of the Casimir element, which belongs to the dual to ${\cal A}_q$ quantum group $U_q(SL_2({\bf C}))$, on some subspace in ${\bf L}_q^3$ in these coordinates leads to a second order difference operator on the infinite one-dimensional lattice. In the continuos limit $q\rightarrow 1$ it is transformed into the Schr\"{o}dinger Hamiltonian, which describes zero modes into the Liouville field theory (the Liouville quantum mechanics). We calculate the spectrum (Brillouin zones) and the eigenfunctions of this operator. They are $q$-continuos Hermit polynomials, which are particular case of the Macdonald or Rogers-Askey-Ismail polynomials. The scattering in this problem corresponds to the scattering of first two level dressed excitations in the $Z_N$ Baxter model in the very peculiar limit when the anisotropy parameter $\ga$ and $N~\rightarrow\infty$, or, equivalently, $(\ga, N)\rightarrow 0$.