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In this paper we study in a Hilbert space a homogeneous linear second order difference equation with nonconstant and noncommuting operator coefficients. We build its exact resolutive formula consisting in the explicit non-iterative expression of a generic term of the unknown sequence of vectors of the Hilbert space. Some non-trivial applications are reported with the aim of showing the usefulness and the broad applicability of our result.
The exact solution of a Cauchy problem related to a linear second-order difference equation with constant noncommutative coefficients is reported.
The detailed construction of the general solution of a second order non-homogenous linear operatordifference equation is presented. The wide applicability of such an equation as well as the usefulness of its resolutive formula is shown by studying so
The singularity structure of a second-order ordinary differential equation with polynomial coefficients often yields the type of solution. If the solution is a special function that is studied in the literature, then the result is more manageable usi
We construct the Baxters operator and the corresponding Baxters equation for a quantum version of the Ablowitz Ladik model. The result is achieved by looking at the quantum analogue of the classical Backlund transformations. For comparison we find th
The method, proposed in the given work, allows the application of well developed standard methods used in quantum mechanics for approximate solution of the systems of ordinary linear differential equations with periodical coefficients.