Application of spectral mapping method to Dirac operator

In the present study, theorems related to the uniqueness of the solution of inverse problems for Dirac equations system are proved by applying spectral mapping method. With the help of this method, the inverse problem is reduced to the so-called main equation, which corresponds to the problem of existence and uniqueness of the solution of the system of linear equations in the Banach space.

PDF

___

[1] Abdukadyrov E. Computation of the regularized trace for a Dirac system. Vestnik Moskov Univercity Serial Mathematics and Mechanics 1967; 22 (4): 17-24.
[2] Amirov RKh. On a system of Dirac differential equations with discontinuity conditions inside an interval. Ukrainian Mathematical Journal 2005; 57 (5): 712-727.
[3] Evans WD, Harris BJ. Bounds for the point spectra of separated Dirac operators. Proceeding Royal Society 1981; 88 (1): 1-15.
[4] Freiling G, Yurko VA. Inverse Sturm-Liouville Problems and Their Applications. New York, NY, USA: Nova Science, 2001.
[5] Gasymov MG, Levitan BM. The inverse problem for the Dirac system. Doklady Akademy Nauk SSSR 1966; 167 (6): 967-970.
[6] Gasymov MG. The inverse scattering problem for a system of Dirac equations of order 2n. Doklady Akademy Nauk SSR 1966; 167 (6): 1219-1222.
[7] Gasymov MG, Dzhabiev TT. On the determination of the Dirac system from two spectra. In: Transactions Of The Summer School On Spectral Theory Operator; Baku, Azerbaijan; 1975. pp. 46-71.
[8] Khasanov AB. On eigenvalues of the Dirac operator located on the continuous spectrum theory of mathematical physics. Theoretical and Mathematical Physics 1994; 99 (1): 20-26.
[9] Levin BYA. Lectures On Entire Functions. Providence, RI, USA: American Mathematical Society, 1996.
[10] Levitan BM, Sargsjan IS. Sturm-Liouville and Dirac Operators. Dordrecht, Holland: Kluwer Academic Publishers Group, 1991.
[11] Maksudov FG, Veliev SG. The inverse scattering problem for the nonselfadjoint Dirac operator on the whole axis. Doklady Akademy Nauk SSSR 1975; 225 (6): 1263-1266.
[12] Marchenko VA. Some problems in the theory of second-order differential operators. Doklady Akademy Nauk SSSR 1950; 72: 457-560.
[13] Nauamura G, Tsuchidat T. Uniqueness for an inverse boundary value problem for Dirac operators. Community Partial Differential Equaitons 2000; 25 (7): 557-577. doi: 10.1080/03605300008821551
[14] Otelbayev MO. Distribution of the eigenvalues of the Dirac operator. Mathematic Zametki 1973; 14 (6): 843-852.
[15] Ozkan AS, Amirov RKh. An interior inverse problem for the impulsive Dirac operator. Tamkang Journal of Mathematics 2011; 42 (3): 259-263.
[16] Panakhov ES. Determination of a Dirac system from two incompletely given sets of eigenvalues. Akademy Nauk Azerbaijan SSSR Doklady 1985; 41 (5): 8-12.
[17] Pöschel J, Trubowitz E. Inverse Spectral Theory. Orlando, FL, USA: Pure Applied Mathematics, 1987.
[18] Sargsjan IS. A theorem of the completeness of the eigenfunctions of the generalized Dirac system. Doklady Akademy Nauk Armian SSSR 1966a; 42 (2): 77-82.
[19] Sargsjan IS. Solution of the Cauchy problem for a one-dimensional Dirac system. Izvesty Akademy Nauk Armian SSSR Serial Mathematics 1966d; 1 (6): 392-436.