The main aim of this paper is to construct quantum extension of the discrete Sturm–Liouville equation consisting of second-order difference equation and boundary conditions that depend on a quadratic eigenvalue parameter. | Turk J Math (2016) 40: 445 – 452 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Quadratic eigenparameter-dependent quantum difference equations Yelda AYGAR∗ Department of Mathematics, Faculty of Science, Ankara University, Ankara, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: The main aim of this paper is to construct quantum extension of the discrete Sturm–Liouville equation consisting of second-order difference equation and boundary conditions that depend on a quadratic eigenvalue parameter. We consider a boundary value problem (BVP) consisting of a second-order quantum difference equation and boundary conditions that depend on the quadratic eigenvalue parameter. We present a condition that guarantees that this BVP has a finite number of eigenvalues and spectral singularities with finite multiplicities. Key words: q -difference equation, Jost solution, spectral analysis, eigenvalue, spectral singularity 1. Introduction Boundary value problems (BVPs) for difference equations have been intensively studied in the last decade. The modelings of certain problems in engineering, economics, control theory, and other areas of study have led to the rapid development of the theory and also spectral theory of difference equations. Some problems of spectral theory for difference equations were treated by various authors [1, 9, 4, 8, 5]. Furthermore, quantum calculus received a lot of attention, and most of the published work has been interested in some problems of q -difference equations (see [2, 6, 3, 11, 10]). The spectral analysis of eigenparameter-dependent nonselfadjoint difference equations and q -difference equation were studied in [7, 15, 6, 13]. A BVP for the discrete Sturm–Liouville equation consisting of second-order difference equation and boundary conditions that depend on a quadratic eigenvalue parameter was first studied by .
