Abstract
This paper is devoted to studying a q-analog of the singular Dirac problem. First, we investigate some spectral properties of the problem. Then we prove the existence of a spectral function and establish a Parseval’s equality, for the singular q-Dirac system in a Hilbert space. Although there were given some results for this type of problem, we think that Parseval’s equality has not been studied yet.
Highlights
In 1910, Jackson introduced the q-derivative operator, Dq, different from the classical derivative, and its right-inverse, the q-integration [19, 20]
The q-calculus was based on these notations. This calculus has a lot of applications in different mathematical areas, such as calculus of variations, orthogonal polynomials, theory of relativity, quantum theory and statistical physics
3 Spectral properties of the q-Dirac problem we investigate some spectral properties of the q-Dirac problem (1.1)–(1.3)
Summary
In 1910, Jackson introduced the q-derivative operator, Dq, different from the classical derivative, and its right-inverse, the q-integration [19, 20]. The q-calculus was based on these notations. This calculus has a lot of applications in different mathematical areas, such as calculus of variations, orthogonal polynomials, theory of relativity, quantum theory and statistical physics (see [1, 23, 29]). There are several physical models involving q-functions, q-derivatives, q-integrals and their related problems [9, 11, 12, 15, 28]. Let us consider the q-problem which consists of the q-Dirac system. Where λ is the spectral parameter, q ∈ (0, 1) is fixed, p(x) and r(x) are real-valued functions and continuous at zero, and p(x), r(x) ∈ L1q(0, ∞)
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