Qes Systems, Invariant Spaces and Polynomials Recursions

Abstract

Let us denote V, the finite dimensional vector spaces of functions of the form ψ(x) = p n (x) + f (x)p m (x) where p n (x) and p m (x) are arbitrary polynomials of degree at most n and m in the variable x while f (x) represents a fixed function of x. Conditions on m, n and f (x) are found such that families of linear differential operators exist which preserve V. A special emphasis is accorded to the cases where the set of differential operators represents the envelopping algebra of some abstract algebra. These operators can be transformed into linear matrix valued differential operators. In the second part, such types of operators are considered and a connection is established between their solutions and series of polynomials-valued vectors obeying three terms recurence relations. When the operator is quasi exactly solvable, it possesses a finite dimensional invariant vector space. We study how this property leads to the truncation of the polynomials series.

Quadratic Lie algebras and quasiexact solvability of the two-photon Rabi Hamiltonian

“…The fact that the generators under discussion are quasi-exactly solvable is by no means obvious and was established within the approach called in [12] QES-extension. It was shown there that the differential operators J ± 2 (26), J ± 3 (29) possess invariant subspaces ℜ 2 N , ℜ 3 N which are not polynomials and the generators cannot be obtained as a polynomial deformation of the linear algebra sl(2, R) [10]. It turned out that they can be expressed in terms of hypergeometric functions [12].…”

Extended Jaynes–Cummings models and (quasi)-exact solvability

“…In the case of standard QES equations [15] there it appears a three terms recurence relations which leads to sets of orthogonal relation. In the case of systems of QES equations adressed in [16] the recurence relation is also three terms but the situation here is quite different. Actually, it is to our knowledge, an open question to know whether the set of polynomials (p j (E), q j (E)) are somehow orthogonal as it is the case for standard scalar equation.…”

Quasi-exactly solvable models based on special functions

“…The operator P 4 ͑24͒ as well as operator P 2 ͑13͒, up to an additive constant, depends on two free parameters d 13 …”

“…12,13 In the present paper we consider the problem of constructing QES operators which preserve subspaces of a more general form, M n = span͕f 1 ͑x͒, f 2 ͑x͒, ... , f n−1 ͑x͒, f n ͑x͖͒. 12,13 In the present paper we consider the problem of constructing QES operators which preserve subspaces of a more general form, M n = span͕f 1 ͑x͒, f 2 ͑x͒, ... , f n−1 ͑x͒, f n ͑x͖͒.…”