Abstract
This work is devoted to get Jost solution by the method of variation of parameters and properties of this Jost solution of a boundary value problem (BVP) depending on an hyperbolic eigenparameter. The results obtained are then used to find the Green function, resolvent and continuous spectrum of this BVP by using the methods of operator theory, methods of functional analysis and the general theory of difference equations. Moreover, using the Weyl compact perturbation theorem, we get that the operator L generated by the q -difference expression has continuous spectrum filling the segment [−2√ q , 2√ q ].
Highlights
Quantum calculus which known as the calculus without limits is a connection between mathematics and physics
boundary value problem (BVP) consisting of q -difference equations and boundary condition depending on spectral parameter is studied in [3, 5]
We discuss the Jost solution and Jost function of of the BVP (1)-(2) and after getting its Green function and Resolvent, we investigate the continuous spectrum of this BVP under the assumption
Summary
Quantum calculus which known as the calculus without limits is a connection between mathematics and physics. The study of q - difference equations has become an important area of research Because such equations have an interest role due to their applications in several mathematical areas such as number theory, orthogonal polynomials, mathematical control theory, basic hyper-geometric functions and other disciplines including mechanic, theory of relativity, biology, economics. Since it has huge applications in several disciplines, spectral analysis of q. Some different aspects on the appearance of Jost solution, resolvent and Green function are stand out This new approach will provide a wide perspective on applications of these problems in physics, mathematics, economics and engineering.
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