Factorization method for second order functional equations

Abstract

We apply general difference calculus in order to obtain solutions to the functional equations of the second order. We show that factorization method can be successfully applied to the functional case. This method is equivariant under the change of variables. Some examples of applications are presented.Comment: 22 pages, examples and new section added, several correction

Factorization method on time scales

“…Let us now consider a sequence of weight functions ρ k : T κ → R + ∪ {0} and corresponding Hilbert spaces H k := L 2 (T, ρ k µ ∆ ) for k in some subset of Z. Following [GO05] we assume that weight functions satisfy a recurrence relation…”

“…The aim of this paper is to reformulate and generalize the results of papers [GO05,GO02] for the case of calculus on time scales introduced in [Hil90]. Namely in those papers a factorization method for second order difference equations of the form α(x)ψ(τ 2 (x)) + β(x)ψ(τ (x)) + γ(x)ψ(x) = 0 (1.1) [Kli93,RLZ03])…”

Discrete Quantum Harmonic Oscillator

“…This type of factorizations was presented in detail for the general situation in a discrete case in [16][17][18] and in [19][20][21] for the τ-and q-cases too. It is based on classical methods taken from the work of some founders of quantum mechanics such as Schrödinger [22] (see also [23][24][25]).…”

Change of Variables in Factorization Method for Second-order Functional Equations

“…The factorization method for the generalized difference equation (functional equation) was presented in [1,2]. This approach admits a change of variables consistent with its structure.…”