Quasi-exactly Solvable (QES) Equations with Multiple Algebraisations

Abstract

We review three examples of quasi exactly solvable (QES) Hamitonians which possess multiple algebraisations. This includes the most prominent example, the Lamé equation, as well as recently studied many-body Hamiltonians with Weierstrass interaction potential and finally, a 2 × 2 coupled channel Hamiltonian.

Quasi-exact solvability in a general polynomial setting

“…For example, the harmonic oscillator possess one algebraic sector corresponding to even eigenfunctions, and another corresponding to odd eigenfunctions. The existence of multiple algebraizations for the Lamé potential has been noted in [1,4] We will say that a second-order operator T possesses multiple SL 2 algebraizations if there exists a local coordinate z such that T preserves both the vector space P n (z) and the vector space φ(z)P n(z). A distinct, second algebraic sector arises if φ(z) is not a polynomial, leading to more algebraic eigenfunctions.…”

“…(M4b) If k i ∈ 1 4 Z for all i, there are four algebraic sectors. (M8) If k i ∈ 1 2 Z for all i, there are eight algebraic sectors. We observe that if the root structure of p(z) is degenerate, i.e.…”

“…Multiple SL 2 algebraizations have been discussed in [1,6,11], but a full classification of potentials admitting multiple SL 2 algebraizations was not known. Here we provide the full classification thanks to the explicit potential form shown in equations ( 18) and (19).…”

“…(M2b) If there is no i such that k i ∈ 1 2 Z, but k i ± k j ∈ 1 2 Z for exactly two choices of i = j , then there are two algebraic sectors. (M4a) If k i ∈ 1 2 Z for exactly two values of i, there are four algebraic sectors. (M4b) If k i ∈ 1 4 Z for all i, there are four algebraic sectors.…”

“…every operator that preserves P n belongs to the enveloping algebra of sl 2 . 1 We should point out that the hydrogen atom potential, and more generally, the class of Natanzon exactly-solvable potentials do not fall directly into this framework, but that they can be recovered in terms of invariant spaces of polynomials through a slightly different approach.…”