QES systems, invariant spaces and polynomials recursions

Let us denote \({\cal V}\), the finite dimensional vector spaces of functions of the form \(\psi(x) = p_n(x) + f(x) p_m(x)\) 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 \({\cal V}\). 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.