[Paper Review] Deformation of two body quantum Calogero-Moser-Sutherland models

Kenji Taniguchi|ArXiv.org|Jul 25, 2006

Nonlinear Waves and Solitons4 references3 citations

TL;DR

This paper investigates deformations of two-body quantum Calogero-Moser-Sutherland models by deriving necessary conditions for the singular locus of the potential function and constructing a new elliptic $B_2$-type model. It establishes constraints on coupling constants and vector configurations, and explicitly constructs a fifth-order commutant for a deformed four-line singular locus, proving complete integrability under specific parameter conditions.

ABSTRACT

The possibility of deformation of two body quantum Calogero-Moser-Sutherland models is studied. Obtained are some necessary conditions for the singular locus of the potential function. Such locus is determined if it consists of two, three or four lines. Furthermore, a new deformation of elliptic $B_{2}$ type Calogero-Moser-Sutherland model is explicitly constructed.

Motivation & Objective

• To classify and construct deformed completely integrable two-body quantum Calogero-Moser-Sutherland models beyond standard root system structures.
• To determine necessary conditions on the singular locus $Σ$ and coupling constants $C_\alpha$ for the existence of non-trivial commutants.
• To explicitly construct a new deformation of the elliptic $B_2$-type Calogero-Moser-Sutherland model with a four-line singular locus.
• To analyze the interplay between the geometry of the singular locus and the algebraic structure of commuting differential operators.
• To extend known results on integrability to non-root system configurations, particularly for $\#\mathcal{H} = 2,3,4$.

Proposed method

• Derives necessary conditions on the singular locus $\mathcal{H}$ and coupling constants $C_\alpha$ via analysis of the principal symbol and commutativity of differential operators.
• Applies symmetry and reflection group arguments to constrain possible configurations of $\mathcal{H}$, especially for $\#\mathcal{H} = 2,3,4$, under the assumption of a constant principal symbol for the commutant $P$.
• Uses Laurent expansion techniques and Weierstrass $\wp$-function identities to reconstruct potential functions from commutativity conditions.
• Constructs a fifth-order differential operator $P$ that commutes with the Schrödinger operator $L$, using combinations of second- and fifth-order derivatives along specific directions $\alpha_\pm$.
• Introduces notation for directional derivatives $\partial_{x,\alpha}$ and operators $L_\pm$, $A_\pm(5)$ to express the commutant $P$ in terms of $u_1, u_2, u_+, u_-$, and invariants $g_2, g_3$ of the Weierstrass elliptic function.
• Verifies commutativity of $P$ and $L$ via direct computation, relying on known results for $L_\pm$ and their five-order commutants.

Experimental results

Research questions

• RQ1 What are the necessary geometric and algebraic conditions on the singular locus $\mathcal{H}$ and coupling constants $C_\alpha$ for a two-body quantum Calogero-Moser-Sutherland model to admit a non-trivial commutant?
• RQ2 Can a deformed Calogero-Moser-Sutherland model with a non-root system singular locus (e.g., four lines not forming a root system) still be completely integrable?
• RQ3 For a singular locus of size four, what specific configuration of vectors and coupling constants leads to a valid, non-trivial commutant?
• RQ4 How can the potential function be reconstructed from the commutativity conditions, particularly in the elliptic case?
• RQ5 What is the explicit form of a fifth-order differential operator that commutes with the Schrödinger operator in the deformed $B_2$-type model?

Key findings

• For $\#\mathcal{H} = 2$, the two vectors in $\mathcal{H}$ must be orthogonal, resulting in a singular locus of type $A_1 \times A_1$.
• For $\#\mathcal{H} = 3$, the configuration is $\{e_1, \pm a e_1 + e_2\}$; if not a positive root system of type $A_2$, then only the known model from [2] exists with coupling constants equal to one for the non-orthogonal pair.
• For $\#\mathcal{H} = 4$, the only possible configuration is $\{e_1, e_2, \pm a e_1 + e_2\}$, and the model is integrable only under specific coupling conditions.
• A new deformed elliptic $B_2$-type Calogero-Moser-Sutherland model is explicitly constructed with singular locus $\mathcal{H} = \{e_1, e_2, \pm a e_1 + e_2\}$, where $a^2 \neq 7/3, 3/7, (13 \pm 4\sqrt{10})/3$, and the potential is expressed in terms of Weierstrass $\wp$-functions.
• The constructed commutant $P$ is of order six and explicitly given in terms of $L_1, L_2, L_\pm$, $A_\pm(5)$, and various differential and potential terms, with $u_+(t) = 6(a^2+1)\wp(t)$, $u_1(t) = (3a^{-2}-1)u_+(2at)/8$, and $u_2(t) = (3a^2-1)u_+(2t)/8$.
• The model is completely integrable: $[L, P] = 0$ is verified via direct computation using the derived identities and the structure of the $\wp$-function.