The Completeness of System of Eigenfunctions of 1D Dirac Operators

Yıl 2015, Cilt: 4 Sayı: 2, 35 - 43, 08.01.2016

Hüseyin Tuna , Murat Çoruh

https://doi.org/10.17100/nevbiltek.211039

Öz

In this paper, nonself-adjoint 1D Dirac operators in Weyl’s limit-circle  case are studied. Using Krein’s theorems, we investigate the completeness of the system of eigenvectors and associated vectors for these operators. 

Anahtar Kelimeler

-

Bir Boyutlu Dirac Operatörlerinin Özfonksiyonlar Sisteminin Tamlığı

Öz

Bu çalışmada Weyl limit çember durumunda kendine eş olmayan bir boyutlu Dirac operatörleri çalışılmıştır. Krein teoremleri kullanılarak, bu operatörlerin öz ve asosye vektörler sisteminin tamlığı araştırıldı

Anahtar Kelimeler

Dissipatif Dirac Operatörleri, Öz ve asosye vektörler sisteminin tamlığı, Krein teoremi

Kaynakça

• B. P. Allahverdiev, Spectral analysis of dissipative Dirac operators with general boundary conditions, J. Math. Anal. Appl., 283, (2003), 287-303.
• B. P. Allahverdiev, Extensions, dilations and functional models of Dirac operators in limit-circle case. Forum Math. 17 (2005), no. 4, 591-611.
• E. Bairamov, E. Ugurlu, The determinants of dissipative Sturm--Liouville operators with transmission conditions, Math. Comput. Model. 53 (2011) 805-813.
• E. Bairamov, E. Ugurlu, Krein's theorems for a Dissipative Boundary Value Transmission Problem, Complex Anal. Oper. Theory, DOI 10.1007/s11785-011-1180-z.
• A.G. Baskakov, A.V. Derbushev and A.O. Shcherbakov, The method of similar operators in the spectral analysis of non-self-adjoint Dirac operators with non-smooth potentials, Izv. Math. 75 (3) (2011), pp. 445-469.
• P. Djakov and B. Mityagin, Unconditional convergence of spectral decompositions of 1D Dirac operators with regular boundary conditions, Indiana Univ. Math. J., 61 (1) (2012), pp. 359-398.
• P. Djakov and B. Mityagin, Equiconvergence of spectral decompositions of 1D Dirac operators with regular boundary conditions, J. Approximation Theory 164 (7) (2012), pp. 879-927.
• P. Djakov and B. Mityagin, Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators, J. Funct. Anal. 263 (8) (2012), pp. 2300-2332.
• P. Djakov and B. Mityagin, Riesz bases consisting of root functions of 1D Dirac operators, Proc. Amer. Math. Soc. 141 (4) (2013), pp. 1361-1375
• I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Amer. Math. Soc., Providence, 1969
• G. Guseinov, Completeness theorem for the dissipative Sturm--Liouville operator, Doga-Tr. J. Math 17 (1993) 48-54.
• G.Sh. Guseinov, and H. Tuncay, The determinants of perturbation connected with a dissipative Sturm-Liouville operator, J. Math. Anal. Appl. 194, (1995) 39-49.
• H. Hasegawa, Bound states of the one dimensional Dirac equation for scalar and vector square- well potentials, Physica E, 59 (2014), 192-201.
• M.G. Krein, On the indeterminate case of the Sturm-Liouville boundary problem in the interval ğ0;1Ş, Izv. Akad. Nauk SSSR Ser. Mat. 16 (4) (1952) 293-324 (in Russian).
• M.G. Krein and A. A Nudelman, On Some Spectral Properties of an Inhomogeneous String with Dissipative Boundary Condition, J. Operator Theory 22 (1989), 369-395
• B. M. Levitan and I. S. Sargsjan. Sturm-Liouville and Dirac operators. Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1991 (translated from the Russian).
• A. A. Lunyov and M. M. Malamud, On the completeness of the root vectors for first order systems.Dokl. Math. (3) 88 (2013), 678-683.
• A. A. Lunyov and M. M. Malamud, On the completeness and Riesz basis property of root subspaces of boundary value problems for first order systems. Application to the Timoshenko beam model. arXiv:1401.2574 (Submitted on 11 Jan 2014).
• A. A. Lunyov, M. M. Malamud, On Spectral Synthesis for Dissipative Dirac Type Operators. Integr. Equ. Oper. Theory May 2014, DOI 10.1007/s00020-014-2154-9.
• M. M. Malamud and L. L. Oridoroga, On the completeness of root subspaces of boundary value problems for first order systems of ordinary differential equations. J. Funct. Anal. 263 (2012), 1939-1980; arXiv:0320048.
• M. A. Naimark, Linear Differential Operators, 2nd edn., 1968,
• Nauka, Moscow, English
• transl. of 1st. edn., ,2 12 , 1969, New York.
• B. W. Roos and W. C. Sangren, Spectra for a pair of singular first order differential equations, Proc. Amer. Math. Soc, 12 (1961), 468-476.
• B. W. Roos and W. C. Sangren, Spectra for a pair of first order differential equations, San Diego, California, General Atomic Report GA 1373, 1960.
• B. W. Roos and W. C. Sangren, Expansions associated with a pair of singular first-order differential equations, J. Math. Phys., 4, (1963), 999-1008.
• B. Thaller, The Dirac Equation (Springer, 1992).
• E. C. Titchmarsh, Some eigenfunction expansion formulae. Proc. London Math. Soc. 3, 11,(1961), 159-168.
• E. C. Titchmarsh, A problem in relativistic quantum mechanics. Proc. London Math. Soc. 3,11,(1961), 169-192.
• E. C. Titchmarsh, On the nature of the spectrum in problems of relativistic quantum mechanics. Quart. J. Math. Oxford Ser. 2, 12, (1961), 227-240.
• E. C. Titchmarsh, On the nature of the spectrum in problems of relativistic quantum mechanics. II. Quart. J. Math. Oxford Ser. 2, 13, (1962), 181-192.2.
• H. Tuna, Completeness of the rootvectors of a dissipative Sturm--Liouville operators on time scales, , Applied Mathematics and Computation , 228,(2014), 108-115 .
• H. Tuna and A. Eryılmaz, Completeness Theorem for Discontinuous Dirac Systems, Differential Equations and Dynamical Systems , (2013), DOI: 10.1007/s12591-013-0194-2
• H. Tuna and A. Eryılmaz, Completeness of the system of root functions of q-Sturm-Liouville operators, Math. Commun. 19(2014), 65-73.
• H. Tuna and A. Eryılmaz, On the completeness of the root vectors of dissipative Dirac operators with transmission conditions, African Diaspora Journal of Mathematics ,Vol. 17,( 2), (2014), 47- 58.
• J. Weidmann, Spectral Theory of Ordinary Differential Operators, Lecture Notes in Mathematics 1258, Springer, Berlin 1987.