Scattering analysis of a quantum impulsive boundary value problem with spectral parameter

Year 2022, Volume: 51 Issue: 1, 142 - 155, 14.02.2022

Yelda Aygar , Güher Gülçehre Özbey

https://doi.org/10.15672/hujms.912015

Cited By: 2

Abstract

We are interested in scattering and spectral analysis of an impulsive boundary value problem (IBVP) generated with a $q$-difference equation with eigenparameter in boundary condition in addition to impulsive conditions. We work on the Jost solution and scattering function of this problem, and by using the scattering solutions, we establish the resolvent operator, continuous spectrum and point spectrum of this problem. Furthermore, we discuss asymptotic behavior of the Jost solution and properties of eigenvalues.Also, we illustrate our results by a detailed example which is the special case of main problem.

Keywords

q-difference equation, impulsive condition, scattering solution, scattering function, eigenvalue, resolvent operator

References

• [1] M. Adıvar and M. Bohner, Spectral analysis of q-difference equations with spectral singularities, Math. Comput. Model. 43, 695-703, 2006.
• [2] Z.S. Agranovich and V.A. Marchenko, The Inverse problem of scattering theory, Pratt Institute Brooklyn, New York, 1963.
• [3] G.E. Andrews, R. Askey, R. Roy, Special functions (No. 71), Cambridge University Press, 1999.
• [4] Y. Aygar and E. Bairamov, Scattering theory of impulsive Sturm–Liouville equation in quantum calculus, Bull. Malays. Math. Sci. Soc. 42, 3247-3259, 2019.
• [5] Y. Aygar and M. Bohner, On the spectrum of eigenparameter-dependent quantum difference equations, Appl. Math. Inf. Sci. 9, 1725–1729, 2015.
• [6] Y. Aygar and M. Bohner, A polynomial-type Jost solution and spectral properties of a self-adjoint quantum-difference operator, Complex Anal. Oper. Theory 10, 1171– 1180, 2016.
• [7] Y. Aygar and M. Bohner, Spectral analysis of a matrix-valued quantum-difference operator, Dyn. Syst. Appl. 25, 29-37, 2016.
• [8] D.D. Bainov and P.S. Simeonov, Systems with impulse effect stability theory and applications, Ellis Horwood Limited, Chichester, 1989.
• [9] D.D. Bainov and P.S. Simeonov, Impulsive differential equations: asymptotic proper- ties of the solutions, World Scientific, 1995.
• [10] E. Bairamov, Y. Aygar and S. Cebesoy, Investigation of spectrum and scattering function of impulsive matrix difference operators, Filomat 33, 1301-1312, 2019.
• [11] E. Bairamov, Y. Aygar and B. Eren, Scattering theory of impulsive Sturm-Liouville equations, Filomat 31, 5401-5409, 2017.
• [12] E. Bairamov, Y. Aygar and D. Karslioglu, Scattering analysis and spectrum of discrete Schrödinger equations with transmission conditions, Filomat 31, 5391-5399, 2017.
• [13] E. Bairamov, Y. Aygar and G.B. Oznur, Scattering Properties of Eigenparameter- Dependent Impulsive Sturm–Liouville Equations, Bull. Malays. Math. Sci. Soc. 43, 2769-2781, 2020.
• [14] E. Bairamov, S. Cebesoy and I. Erdal, Difference equations with a point interaction, Math. Meth. Appl. Sci. 42, 5498-5508, 2019.
• [15] M. Benchohra, J. Henderson and S. Ntouyas, Impulsive differential equations and inclusions, New York, Hindawi Publishing Corporation, 2006.
• [16] M. Bohner and S. Cebesoy, Spectral analysis of an impulsive quantum difference op- erator, Math. Meth. Appl. Sci. 42, 5331-5339, 2019.
• [17] K.M. Case, On discrete inverse scattering problems II, J. Math. Phys. 14, 916-920, 1973.
• [18] K.M. Case and S.C. Chiu, The discrete version of the Marchenko equations in the inverse scattering problem, J. Math. Phys. 14, 1643-1647, 1973.
• [19] K. Chadon and P.C. Sabatier, Inverse problems in quantum scattering theory, Springer-Verlag, Berlin, New York, 1997.
• [20] İ. Erdal and Ş. Yardımcı, Eigenvalues and scattering properties of difference operators with impulsive condition, Commun. Fac. Sci. Univ. Ank.-Ser, A1 Math. Stat. 68, 663- 671, 2019.
• [21] A. Ergün, A half-inverse problem for the singular diffusion operator with jump con- ditions, Miskolc Math. Notes 21, 805-821, 2020.
• [22] K. Ey, A. Ruffing and S. Suslov, Method of separation of the variables for basic analogs of equations of mathematical physics, Ramanujan J. 13, 407-447, 2007.
• [23] L.D. Faddeev, The construction of the resolvent of the Schrödinger operator for a three-particle system, and the Scattering Problem, Soviet Physics Doklady 7, 600- 602, 1963.
• [24] G. Gasper, M. Rahman and G. George, Basic hypergeometric series, Vol. 96, Cam- bridge University Press, 2004.
• [25] M.G. Gasymov and B. M. Levitan, Determination of the Dirac system from scattering phase, Soviet Mathematics Doklady 67, 1219-1222, 1966.
• [26] R.K. George, A.K. Nandakumaran and A. Arapostathis, A note on controllability of impulsive systems, J. Math. Anal. Appl. 241, 276-283, 2000.
• [27] I.M. Glazman, Direct methods of qualitative spectral analysis of singular Differential Operators, Jerusalem, Israel Program for Scientific Translations, 1965.
• [28] G.S. Guseinov, The inverse problem of scattering theory for a second order difference equation, Soviet Mathematics Doklady 230, 1045-1048, 1976.
• [29] G.S. Guseinov, The determination of an infinite Jacobi matrix from the scattering data, Dokl. Akad. Nauk SSSR 227, 1289-1292, 1976.
• [30] G.S. Guseinov, Boundary value problems for nonlinear impulsive Hamiltonian sys- tems, J. Comput. Appl. Math. 259, 780-789, 2014.
• [31] Y. Khalili and D. Baleanu, Recovering differential pencils with spectral boundary con- ditions and spectral jump conditions, J. Inequal. Appl. 2020, Article No. 262, 2020.
• [32] V. Lakshmikantham and P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, 1989.
• [33] L.A. Lyusternik and V.Y. Sobolev, Elements of Functional Analysis, New York, Hal- sted Press, 1974.
• [34] O.S. Mukhtarov and K. Aydemir, Eigenfunction expansion for Sturm-Liouville prob- lems with transmission conditions at one interior point, Acta Math. Sci. 35, 639-649, 2015.
• [35] O. Mukhtarov, H. Olğar and K. Aydemir, Eigenvalue problems with interface condi- tions, Konuralp J. Math 8, 284-286, 2020.
• [36] M.A. Naimark, Investigation of the Spectrum and the Expansion in Eigenfunction of a Non-selfadjoint Operator of Second Order on a Semi-Axis, Tr. Mosk. Mat. Obs. 3, 181–270, 1954.
• [37] A.M. Samoilenko and N.A. Perestyuk, Impulsive Differential Equations, World Sci- entific, 1995.
• [38] M.R. Ubriaco, Time evolution in quantum mechanics on the quantum line, Phys. Lett. A 163, 1-4, 1992.