Difference ladder operators for a harmonic Schrödinger oscillator using unitary linear lattices

Square integrability of Gaussian bells on time scales

“…For a quadratic potential V(x) -~ x 2, occurring, e.g., in harmonic oscillator interactions, the eigenvalue problem (1) can be solved by the following ladder formalism: decompose the Hamiltonian 7-/into ~/=AtA+~ --~xx+X ~xx+X +Z,…”

“…Now, keeping in mind the decomposition (2) The corresponding eigenvalues are En = )% + 1 = 2n + 1. It can be shown that the eigenfunctions ~n constitute an orthogonal basis of L2(R)--indicating that every probability density I¢12 solving (1) in this case can be written in terms of the normalized eigenfunctions ¢~: oo ¢(x) = E coco(x). n=O The physical interpretation of this is that the coefficients c~ of the state ¢ show that the probability of measuring the particle at energy level En = 2n + 1 is Ic~l 2.…”

“…It is essential that this function be square integrable, since it must be possible to normalize its square to a probability density. Similar ladder formalisms have been constructed in order to consider discrete generalizations of this Schr5dinger scenario, e.g., on so-called unitary linear lattices, being a certain mixture of q-linear grids and equidistant lattices--see [1] for details. In any case, it is of interest to find such discrete formalisms, since many potentials V(x) in quantum mechanics do not allow an analytic treatment; for instance V(x) = x 4 cannot even be handled by perturbation methods.…”

Comparison theorems of Hille–Wintner type for dynamic equations on time scales

“…A time scale that is important in the theory of orthogonal polynomials and quantum theory (cf., e.g., [14], [19]…”

Factorization method on time scales

“…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])…”