A Recursive Construction of Joint Eigenfunctions for the Hyperbolic Nonrelativistic Calogero-Moser Hamiltonians

Abstract

We obtain symmetric joint eigenfunctions for the commuting partial differential operators associated to the hyperbolic Calogero-Moser N-particle system. The eigenfunctions are constructed via a recursion scheme, which leads to representations by multidimensional integrals whose integrands are elementary functions. We also tie in these eigenfunctions with the Heckman-Opdam hypergeometric function for the root system A N−1 .

SQED3 and SQCD3 : Phase transitions and integrability

“…Using recent work on the construction, by a recursive method, of the joint eigenfunctions of this integrable system [20], we show now that the Abelian theory above can be identified with this two-particle A 1 hyperbolic Calogero-Moser, where the coupling constant g in Eq. (8) will be identified with the half-number of flavors N. In particular, this two-particle interpretation follows from considering the function Ψ 2 ðg; x; yÞ ≡ e iy 2 ðx…”

“…The appearance of the quantum mechanical interpretation with a solvable Pöschl-Teller potential immediately suggests a possible role of the hyperbolic Calogero-Moser model, the celebrated integrable system, which can be seen as the many-body generalization of the quantum mechanical problem above. The Hamiltonian of the AN −1 hyperbolic Calogero-Moser model is [19,20]…”

Product formulas for the relativistic and nonrelativistic conical functions

“…From the proof of Prop. 4.3 in[HR15] we then deduce (4.7). Finally, to show (4.8), we first write e ηk J(π, β, βg; r, βk) , β; (t + δr − iβg)/2) G(π, β; (t + δr + iβg)/2) exp(i(t − iη)k).…”

“…It equals the function F (g; r, 2k) defined by Eq. (65) of our joint paper [HR15]. Its relation to the conical (or Mehler) function P 1/2−g ik−1/2 (cosh r) is given by…”

Series Solutions of the Non-Stationary Heun Equation

“…However, in the present case, the second of these equations is trivially solved by E = E (0) n , and this implies a stronger result: Theorem 6.3. Let n ∈ Z, −λ / ∈ N 0 for n > 0 and −(g 0 + g 1 ) / ∈ N 0 for n < 0, f ( ) m (z) m∈Z the functions defined and characterized in Lemma 2.2, and assume that (κ) = 0 and g 0 + g 1 ∈ R. 11 Then the non-stationary Heun equation in (1.2) has a unique solution ψ n (x), E n as in (1.5)-(1.6) given by (3.2), (5.3), (5.13) and the coefficients…”