Matrix Valued Difference Equations with Spectral Singularities
In this study, we examine some spectral properties
of non-selfadjoint matrix-valued difference equations consisting of
a polynomial-type Jost solution. The aim of this study is to
investigate the eigenvalues and spectral singularities of the difference
operator L which is expressed by the above-mentioned difference
equation. Firstly, thanks to the representation of polynomial type Jost
solution of this equation, we obtain asymptotics and some analytical
properties. Then, using the uniqueness theorems of analytic functions,
we guarantee that the operator L has a finite number of eigenvalues
and spectral singularities.
[1] M. A. Naimark, Investigation of the spectrum and the expansion in
eigenfunctions of a non-selfadjoit operators of second order on a
semi-axis. AMS Transl. (2), 16 (1960), 103-193.
[2] V. A. Marchenko, Sturm-Liouville Operators and Applications,
Birkhauser Verlag, Basel, (1986).
[3] J. T. Schwartz, Some non-self adjoint operators, Comm. Pure Appl. Math.,
13 (1960), 609-639.
[4] B. Nagy, Operators with spectral singularities, J. Oper. Theory, 15 (1986),
2, 307-325.
[5] V. E. Lyance, A differential operator with spectral singularities, I, II, AMS
Transl. (2), 60 (1967), 185-225, 227-283.
[6] B. S. Pavlov, The non-selfadjoint Schr¨odinger operator, Topics in Math
Phys. 1, (1967), 87-110.
[7] A. M. Krall, The adjoint of a differential operator with integral boundary
conditions, Proc. Amer. Math. Soc., 16 (1965), 738-742.
[8] E. Bairamov, O. Cakar and A.M. Krall, An eigenfunction expansion for
a quadratic pencil of a Schr¨odinger operator with spectral singularities,
J. Differential Equations 151 (1999), 268-289.
[9] E. Bairamov, O. Cakar and C. Yanık, Spectral singularities Klein-Gordon
s-wave equation, Indian J. Pure Appl. Math., 32 (2001), 851-857.
[10] A. M. Krall, E. Bairamov and O. Cakar, Spectrum and spectral
singularities of a quadratic pencil of a Schr¨odinger operator with a general
boundary condition. J. Differential Equations. 151 (1999), 2, 252-267.
[11] E. Bairamov and A. M. Krall, Dissipative operators generated by Sturm-
Liouville differential expression in the Weyl limit circle case, J. Math.
Anal. Appl., 254 (2001), 178-190.
[12] E. Bairamov and O. Karaman, Spectral singularities of Klein- Gordon
s-wave equations with an integral boundary condition, Acta Math.
Hungar., 97 (2002), 1-2, 121-131.
[13] A. Mostafazadeh, H. Mehri-Dehnavi, Spectral singularities,
biorthonotmal systems and a two-parameter family of complex
point interactions, J. Phys. A. Math. Theory, 42 (2009), 125303 (27 pp).
[14] A. Mostafazadeh, Spectral singularities of a general point interaction, J.
Phys. A. Math. Theory, 44 (2011), 375302 (9 pp).
[15] A. Mostafazadeh, Optical spectral singularities as threshold resonances,
Physical Review A, 83, 045801 (2011).
[16] A. Mostafazadeh, Resonance phenomenon related to spectral
singularities, complex barrier potential, and resonating waveguides,
Physical Review A, 80, 032711 (2009).
[17] A. Mostafazadeh, Spectral singularities of complex scattering potentials
and infinite reflection and transmission coefficients at real energies,
Physical Review Letters, 102 , 220402 (2009).
[18] R. P. Agarwal, Difference Equation and Inequalities, Theory, Methods
and Applications, Marcel Dekker, New York, 2000.
[19] R. P. Agarwal, P. J. Wong, Advanced Topics in Difference Equations,
Kluwer, Dordrecht, 1997.
[20] E. Bairamov, O. C¸ akar, A. M. Krall, Non-self adjoint difference
operators and Jacobi matrices with spectral singularities, Math. Nachr.229
(2001), 5-14. [21] A. M. Krall, E. Bairamov and O. Cakar, Spectral analysis of a
non-selfadjoint discrete Schr¨odinger operators with spectral singularities,
Math. Nachr., 231 (2001), 89-104.
[22] M. Adıvar and E. Bairamov, Spectral properties of non-self adjoint
difference operators. J. Math. Anal. Appl., 261(2), (2001), 461-478.
[23] S. Yardımcı, A note on the spectral singularities of non-self adjoint
matrix-valued difference operators, J. Comput. Appl. Math. 234,(2010),
3039-3042.
[24] Y. Aygar and E. Bairamov, Jost solution and the spectral properties of
the matrix-valued difference operators. Appl. Math. Comput., 218(19),
(2012), 9676-9681.
[25] Lazar’ A. Lusternik and Vladimir J. Sobolev, Elements of functional
analysis. Hindistan Publishing Corp., Delhi, russian edition, 1974.
International Monographs on Advanced Mathematics&Physics.
[26] V.P. Serebryakov, An inverse problem of scattering theory for difference
equations with matrix coefficients, Dokl. Akad. Nauk. SSSR 250,
562-565, (1980).
[27] I. M. Glazman, Direct methods of qualitative spectral analysis of singular
differential operators. Translated from the Russian by the IPST staff.
[28] E. P. Dolzhenko, Boundary value uniqueness theorems for analytic
functions, Math. Notes,25 , (1979), 437-442.
[1] M. A. Naimark, Investigation of the spectrum and the expansion in
eigenfunctions of a non-selfadjoit operators of second order on a
semi-axis. AMS Transl. (2), 16 (1960), 103-193.
[2] V. A. Marchenko, Sturm-Liouville Operators and Applications,
Birkhauser Verlag, Basel, (1986).
[3] J. T. Schwartz, Some non-self adjoint operators, Comm. Pure Appl. Math.,
13 (1960), 609-639.
[4] B. Nagy, Operators with spectral singularities, J. Oper. Theory, 15 (1986),
2, 307-325.
[5] V. E. Lyance, A differential operator with spectral singularities, I, II, AMS
Transl. (2), 60 (1967), 185-225, 227-283.
[6] B. S. Pavlov, The non-selfadjoint Schr¨odinger operator, Topics in Math
Phys. 1, (1967), 87-110.
[7] A. M. Krall, The adjoint of a differential operator with integral boundary
conditions, Proc. Amer. Math. Soc., 16 (1965), 738-742.
[8] E. Bairamov, O. Cakar and A.M. Krall, An eigenfunction expansion for
a quadratic pencil of a Schr¨odinger operator with spectral singularities,
J. Differential Equations 151 (1999), 268-289.
[9] E. Bairamov, O. Cakar and C. Yanık, Spectral singularities Klein-Gordon
s-wave equation, Indian J. Pure Appl. Math., 32 (2001), 851-857.
[10] A. M. Krall, E. Bairamov and O. Cakar, Spectrum and spectral
singularities of a quadratic pencil of a Schr¨odinger operator with a general
boundary condition. J. Differential Equations. 151 (1999), 2, 252-267.
[11] E. Bairamov and A. M. Krall, Dissipative operators generated by Sturm-
Liouville differential expression in the Weyl limit circle case, J. Math.
Anal. Appl., 254 (2001), 178-190.
[12] E. Bairamov and O. Karaman, Spectral singularities of Klein- Gordon
s-wave equations with an integral boundary condition, Acta Math.
Hungar., 97 (2002), 1-2, 121-131.
[13] A. Mostafazadeh, H. Mehri-Dehnavi, Spectral singularities,
biorthonotmal systems and a two-parameter family of complex
point interactions, J. Phys. A. Math. Theory, 42 (2009), 125303 (27 pp).
[14] A. Mostafazadeh, Spectral singularities of a general point interaction, J.
Phys. A. Math. Theory, 44 (2011), 375302 (9 pp).
[15] A. Mostafazadeh, Optical spectral singularities as threshold resonances,
Physical Review A, 83, 045801 (2011).
[16] A. Mostafazadeh, Resonance phenomenon related to spectral
singularities, complex barrier potential, and resonating waveguides,
Physical Review A, 80, 032711 (2009).
[17] A. Mostafazadeh, Spectral singularities of complex scattering potentials
and infinite reflection and transmission coefficients at real energies,
Physical Review Letters, 102 , 220402 (2009).
[18] R. P. Agarwal, Difference Equation and Inequalities, Theory, Methods
and Applications, Marcel Dekker, New York, 2000.
[19] R. P. Agarwal, P. J. Wong, Advanced Topics in Difference Equations,
Kluwer, Dordrecht, 1997.
[20] E. Bairamov, O. C¸ akar, A. M. Krall, Non-self adjoint difference
operators and Jacobi matrices with spectral singularities, Math. Nachr.229
(2001), 5-14. [21] A. M. Krall, E. Bairamov and O. Cakar, Spectral analysis of a
non-selfadjoint discrete Schr¨odinger operators with spectral singularities,
Math. Nachr., 231 (2001), 89-104.
[22] M. Adıvar and E. Bairamov, Spectral properties of non-self adjoint
difference operators. J. Math. Anal. Appl., 261(2), (2001), 461-478.
[23] S. Yardımcı, A note on the spectral singularities of non-self adjoint
matrix-valued difference operators, J. Comput. Appl. Math. 234,(2010),
3039-3042.
[24] Y. Aygar and E. Bairamov, Jost solution and the spectral properties of
the matrix-valued difference operators. Appl. Math. Comput., 218(19),
(2012), 9676-9681.
[25] Lazar’ A. Lusternik and Vladimir J. Sobolev, Elements of functional
analysis. Hindistan Publishing Corp., Delhi, russian edition, 1974.
International Monographs on Advanced Mathematics&Physics.
[26] V.P. Serebryakov, An inverse problem of scattering theory for difference
equations with matrix coefficients, Dokl. Akad. Nauk. SSSR 250,
562-565, (1980).
[27] I. M. Glazman, Direct methods of qualitative spectral analysis of singular
differential operators. Translated from the Russian by the IPST staff.
[28] E. P. Dolzhenko, Boundary value uniqueness theorems for analytic
functions, Math. Notes,25 , (1979), 437-442.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:71190", author = "Serifenur Cebesoy and Yelda Aygar and Elgiz Bairamov", title = "Matrix Valued Difference Equations with Spectral Singularities", abstract = "In this study, we examine some spectral properties
of non-selfadjoint matrix-valued difference equations consisting of
a polynomial-type Jost solution. The aim of this study is to
investigate the eigenvalues and spectral singularities of the difference
operator L which is expressed by the above-mentioned difference
equation. Firstly, thanks to the representation of polynomial type Jost
solution of this equation, we obtain asymptotics and some analytical
properties. Then, using the uniqueness theorems of analytic functions,
we guarantee that the operator L has a finite number of eigenvalues
and spectral singularities.
", keywords = "Difference Equations, Jost Functions, Asymptotics,
Eigenvalues, Continuous Spectrum, Spectral Singularities.", volume = "9", number = "11", pages = "647-4", }
