Abstract
Spectral analysis of a boundary value problem (BVP) consisting of a second-order quantum difference equation and boundary conditions depending on an eigenvalue parameter with spectral singularities was first studied by Aygar and Bohner (Appl. Math. Inf. Sci. 9(4):1725-1729, 2015). The main goal of this paper is to construct the principal vectors corresponding to the eigenvalues and the spectral singularities of this BVP. These vectors are important to get the spectral expansion formula for this BVP.
Highlights
Many areas including mathematical physics, engineering, economics, and quantum mechanics need the spectrum of differential and discrete operators to solve some problems
Let us consider the boundary value problem (BVP) consisting of the second-order q-difference equation tt qa(t)y(qt) + b(t)y(t) + a y
In Section, we obtain principal vectors corresponding to eigenvalues and spectral singularities of L, and give some properties of them
Summary
Many areas including mathematical physics, engineering, economics, and quantum mechanics need the spectrum of differential and discrete operators to solve some problems. Many authors have investigated the spectral analysis of differential and discrete operators [ – ]. In addition to differential and discrete equations, the spectral theory of quantum difference equations has been treated in the last decade [ – ]. In [ ], it is proved that the operator L has a finite number of eigenvalues and spectral singularities with finite multiplicities under the condition ln t δ sup exp ε. In Section , we obtain principal vectors corresponding to eigenvalues and spectral singularities of L, and give some properties of them. This paper will be valuable for readers because principal vectors that we obtained corresponding to the eigenvalues and spectral singularities are important to find the spectral expansion of the operator L. Let us define the function f using
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