2006/01/02 by Nobuyuki Motoyui, Y. Brihaye, Yves Brihaye +5 · 6 citations
Computer Science · Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebra over a field #Differential operator #Invariant (physics) #Mathematical physics #Mathematics #Matrix Theory and Algorithms #Nonlinear Waves and Solitons #Operator (biology) #Physics #Pure mathematics #Vector space #hep-th #math-ph #math.MP
paper · pdf · open access · doi:10.1142/s0217732307023912
published in Modern Physics Letters A 22(19), 1423-1438 (World Scientific) · 18 pages, no figures
arxiv created 2006/01/02 · openalex publication_date 2007/06/21 · arxiv updated 2016/12/21 · openalex created_date 2019/06/27 · openalex updated_date 2026/08/05
Let us denote [Formula: see text], 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 [Formula: see text]. A special emphasis is accorded to the cases where the set of differential operators represents the enveloping 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 recurrence 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.