Inverse scattering properties with Levinson formula for cubic eigenparameter-dependent discrete Dirac problem

Turhan Koprubasi; R. N. Mohapatra

doi:10.3842/umzh.v76i9.7718

Turhan Koprubasi Department of Mathematics, Kastamonu University, Turkey and Department of Mathematics, University of Central Florida, Orlando, FL, USA
R. N. Mohapatra Department of Mathematics, University of Central Florida, Orlando, FL, USA

DOI: https://doi.org/10.3842/umzh.v76i9.7718

Keywords: Discrete Dirac equation, Eigenparameter, Scattering function, Levinson formula

Abstract

UDC 517.9

We present some properties of zeros of the Jost function and of the scattering function. Then the uniqueness of the kernel and the continuity of the scattering function with appropriate Levinson-type formula are investigated for the inverse discrete Dirac problem based on the cubic eigenparameter-dependent boundary condition.

References

R. P. Agarwal, Difference equation and inequalities: theory, methods and applications, Marcel Dekker Inc., New York, Basel (2000).

Z. S. Agranovich, V. A. Marchenko, The inverse problem of scattering theory, Pratt Institute Brooklyn, New York (1963).

K. Chadan, P. C. Sabatier, Inverse problems in quantum scattering theory, Springer-Verlag, Berlin (1989); https://doi.org/ 10.1007/978-3-642-83317-5.

W. G. Kelley, A. C. Peterson, Difference equations: an introduction with applications, Academic Press, San Diego (2001).

N. Levinson, On the uniqueness of the potential in a Schrödinger equation for a given asymptotic phase, Mat.-Fys. Meddelelser, 25, № 9, 1–30 (1949).

B. M. Levitan, I. S. Sargsyan, Sturm–Liouville and Dirac operators, Kluwer, Dordrecht (1991); https://doi.org/ 10.1007/978-94-011-3748-5.

M. A. Naimark, Linear differential operators, part I, II: Linear differential operators in Hilbert space, Frederick Ungar Publ. Co., London (1968).

E. C. Titchmarsh, Introduction to the theory of Fourier integral, Oxford Univ. Press, London (1948).

A. Vretblad, Fourier analysis and its applications, Springer-Verlag, New York (2003).

O. Akin, E. Bairamov, On the structure of discrete spectrum of the non-selfadjoint system of differential equations in the first order, J. Korean Math. Soc., 32, № 3, 401–413 (1995).

M. Adivar, E. Bairamov, Spectral properties of non-selfadjoint difference operators, J. Math. Anal. and Appl., 261, № 2, 461–478 (2001); https://doi.org/10.1006/jmaa.2001.7532.

Y. Aygar, M. Olgun, Investigation of the spectrum and the Jost solutions of discrete Dirac system on the whole axis, J. Inequal. and Appl., 73 (2014); https://doi.org/10.1186/1029-242X-2014-73.

E. Bairamov, A. O. Celebi, Spectrum and spectral expansion for the non-selfadjoint discrete Dirac operators, Quart. J. Math. Oxford, 50, № 200, 371–384 (1999); https://doi.org/10.1093/qjmath/50.200.371.

E. Bairamov, C. Coskun, Jost solutions and the spectrum of the system of difference equations, Appl. Math. Lett., 17, № 9, 1039–1045 (2004); https://doi.org/10.1016/j.aml.2004.07.006.

E. Bairamov, S. Solmaz, Spectrum and scattering function of the impulsive discrete Dirac systems, Turk. J. Math., 42, № 6, 3182–3194 (2018); https://doi.org/10.3906/mat-1806-5.

I. M. Guseinov, On the representation of Jost solutions for Dirac's equation system with discontinuous coefficients, Izv. Akad. Nauk Azerb. SSR, 5, 41–45 (1999).

T. Koprubasi, The cubic eigenparameter dependent discrete Dirac equations with principal functions, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. and Stat., 68, № 2, 1742–1760 (2019); https://doi.org/10.31801/cfsuasmas.454232.

T. Koprubasi, R. N. Mohapatra, Spectral analysis of discrete Dirac equation with generalized eigenparameter in boundary condition, Filomat, 33, № 18, 6039–6054 (2019); https://doi.org/10.2298/FIL1918039K.

S. Solmaz, E. Bairamov, Scattering properties of impulsive difference Dirac equations, Turk. J. Math., 46, № 1, 397–405 (2022); https://doi.org/10.3906/mat-2105-70.

R. K. Amirov, B. Keskin, A. S. Ozkan, Direct and inverse problems for the Dirac operator with a spectral parameter linearly contained in a boundary condition, Ukr. Math. J., 61, № 9, 1365–1379 (2009); https://doi.org/10.1007/s11253-010-0282-1.

G. M. Azimova, I. M. Guseinov, Direct and inverse problems of scattering theory for a system of first order difference equations (in Russian), Izv. Akad. Nauk Azerb. SSR, 8, № 3, 3–8 (1987).

G. Bascanbaz Tunca, E. Kir Arpat, Uniqueness of the solution to the inverse problem of scattering theory for the Sturm–Liouville operator system with a spectral parameter in the boundary condition, Gazi Univ. J. Sci., 29, № 1, 135–142 (2016); https://doi.org/10.1155/S0161171204203088.

J. Blackledge, B. Babajanov, On the Dirac scattering problems, Math. Aeterna, 3, № 7, 535–544 (2013); https://doi.org/10.21427/D7JP6B.

A. Çöl, Inverse spectral problem for Dirac operator with discontinuous coefficient and polynomials in boundary condition, Inverse Probl. Sci. and Eng., 24, № 2, 234–246 (2016); https://doi.org/10.1080/17415977.2015.1017487.

B. Frıtzsche, B. Kirstein, I. Y. Roitberg, A. L. Sakhnovich, Discrete Dirac system: rectangular Weyl functions, direct and inverse problems, Operators and Matrices, 8, № 3, 799–819 (2014); https://doi.org/10.7153/oam-08-45.

H. M. Huseynov, A. K. Khanmamedov, R. I. Aleskerov, The inverse scattering problem for a discrete Dirac system on the whole axis, J. Inverse and Ill-Possed Probl., 25, № 6, 829–834 (2017); https://doi.org/10.1515/jiip-2017-0018.

A. K. Khanmamedov, Inverse scattering problem for the difference Dirac operator on a half-line, Dokl. Math., 79, № 1, 103–104 (2009); https://doi.org/10.1134/S1064562409010311.

Kh. R. Mamedov, A. Çöl, On the expansion formula for a class of Dirac operator with discontinuous coefficient, Int. J. Comp. Cog., 7, № 4, 20–24 (2009).

V. A. Zolotarev, Direct and inverse spectral problems for a Dirac operator with non-local potential, J. Math. Anal. and Appl., 503, № 1, Article~125075 (2021); https://doi.org/10.1016/j.jmaa.2021.125075.