tailieunhanh - Dirac systems with regular and singular transmission effects

In this paper, we investigate the spectral properties of singular eigenparameter dependent dissipative problems in Weyl’s limit-circle case with finite transmission conditions. In particular, these transmission conditions are assumed to be regular and singular. | Turk J Math (2017) 41: 193 – 210 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Dirac systems with regular and singular transmission effects ∗ ˘ Ekin UGURLU Department of Mathematics, Faculty of Arts and Science, C ¸ ankaya University, Ankara, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: In this paper, we investigate the spectral properties of singular eigenparameter dependent dissipative problems in Weyl’s limit-circle case with finite transmission conditions. In particular, these transmission conditions are assumed to be regular and singular. To analyze these problems we construct suitable Hilbert spaces with special inner products and linear operators associated with these problems. Using the equivalence of the Lax–Phillips scattering function and Sz-Nagy–Foia¸s characteristic functions we prove that all root vectors of these dissipative operators are complete in Hilbert spaces. Key words: Dissipative operator, first-order system, transmission condition, scattering function, characteristic function 1. Introduction As is known, Dirac systems are of the form y2′ + p(x)y1 + r(x)y2 = λy1 , −y1′ + r(x)y1 + q(x)y2 = λy2 , (1) where λ is a complex parameter, and p, q , and r are real-valued and locally integrable functions on some interval (a, b) ⊆ R. The system (1) plays a central role in relativistic quantum theory. In fact, the system (1) corresponds to Dirac’s radial relativistic wave equation for a particle in a central field [12,14]. One of the important problems of the system (1) is to describe the solutions belonging to squarely integrable space on some singular intervals, that is, intervals in which at least one of the potentials p, q , and r increase boundedlessly. In 1910, Weyl showed with his extraordinary method that at least one of the linearly independent solutions of a singular second-order differential .