Spectral Analysis of Non-selfadjoint Second Order Difference Equation with Operator Coefficient

Gökhan Mutlu; Esra Kır Arpat

doi:10.16984/saufenbilder.627496

Araştırma Makalesi

Yıl 2020, Cilt: 24 Sayı: 3, 494 - 500, 01.06.2020

Gökhan Mutlu Esra Kır Arpat

https://doi.org/10.16984/saufenbilder.627496

Cited By: 4

Öz

Kaynakça

[1] M.A. Naimark, “Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint operator of second order on a semi-axis,” AMS Transl. 2 (16), pp. 103-193, 1960.
[2] M.A. Naimark, “Linear Differential Operators, II,” Ungar, New York, 1968.
[3] V.E. Lyance, “A differential operator with spectral singularities, I-II,” AMS Transl. 2 (60), pp. 185-225, 227-283, 1967.
[4] B.S. Pavlov, “The non-selfadjoint Schrödinger operator in Spectral Theory and Wave Processes, Topics in Mathematical Physics, Vol.1, Birman M.S. (eds), Springer, Boston, MA, pp. 87- 114, 1967.
[5] J.T. Schwartz, “Some non-selfadjoint operators,” Comm. Pure Appl. Math., 13, pp. 609-639, 1960.
[6] C.M. Bender, “Making sense of non- Hermitian Hamiltonians,” Rep. Prog. Phys., 70, pp. 947-1018, 2007.
[7] A. Mostafazadeh, “Pseudo-Hermitian representation of quantum mechanics,” arXiv:0810.5643, 2008.
[8] B.F. Samsonov, “SUSY transformations between digonalizable and nondiagonalizable Hamiltonians,” arXiv:quant-ph/0503075, 2005.
[9] A. Mostafazadeh, “Spectral singularities of complex scattering potentials and infinite reaction and transmission coefficients at real energies,” arXiv:0901.4472, 2009.
[10] R.P. Agarwal, “Difference Equations and Inequalities: Theory, Methods and Applications,” Marcel Dekker, New York, 2000.
[11] R.P. Agarwal and P.J.Y. Wong, “Advanced Topics in Difference Equations,” Kluwer, Dordrecht, 1997.
[12] Y. Aygar and E. Bairamov, “Jost solution and the spectral properties of the matrixvalued difference operators,” Appl. Math. Comput., 218, pp. 9676-9681, 2012.
[13] E. Bairamov, Y. Aygar and S. Cebesoy, “Spectral analysis of a selfadjoint matrix valued discrete operator on the whole axis,” J. Nonlinear Sci. Appl., 9, pp. 4257- 4262, 2016.
[14] R. Carlson, “An inverse problem for the matrix Schrödinger equation,” J. Math. Anal. Appl., 267, pp. 564-575, 2002.
[15] S. Clark, Gesztesy and W. Renger, “Trace formulas and Borg-type theorems for matrix valued Jacobi and Dirac finite difference operators,” J. Differential Equations, 219, pp. 144-182, 2005.
[16] F. Gesztesy, A. Kiselev and K.A. Makarov, “Uniqueness results for matrixvalued Schrödinger, Jacobi and Dirac-type operators,” Math. Nachr., 239, pp. 103- 145, 2002.
[17] E. Bairamov, Ö. Çakar and A.M. Krall, “Non-selfadjoint difference operators and Jacobi matrices with spectral singularities,” Math. Nachr., 229, pp. 5- 14, 2001.
[18] A.M. Krall, E.Bairamov and Ö. Çakar, “Spectral analysis of a non-selfadjoint discrete Schrödinger operators with spectral singularities,” Math. Nachr., 231, pp. 89-104, 2001.
[19] T. Koprubasi and R.N. Mohapatra, “Spectral properties of generalized eigenparameter dependent discrete Sturm- Liouville type equation,” Quaestiones Mathematicae, 40 (4), pp. 491-505, 2017.
[20] M. Adivar and E. Bairamov, “Spectral properties of non-selfadjoint difference operators,” J. Math. Anal. Appl., 261, pp. 461-478, 2001.
[21} M. Adivar and E. Bairamov, “Difference equations of second order with spectral singularities,” J. Math. Anal. Appl., 277, pp. 714-721, 2003.
[22] S. Cebesoy, Y. Aygar and E. Bairamov, “Matrix valued difference equations with spectral singularities,” International Journal of Mathematical, Computational Physical, Electrical and Computer Engineering, 9 (11), pp. 579-582, 2015.
[23] M.V. Keldysh, “On the completeness of the eigenfunctions of some classes of nonselfadjoint linear operators,” Russ. Math. Surv., 26 (4), pp. 15-44, 1971.
[24] V.P. Serebrjakov, “An inverse problem of the scattering theory for difference equations with matrix coeffcients,” Dokl. Akad. Nauk SSSR, 250 (3), pp. 562-565, 1980 (in Russian).
[25] L.A. Lusternik and V.I. Sobolev, “Elements of Functional Analysis,” Halsted Press, New York, 1974.
[26] I.M. Glazman, “Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators,” Israel Program for Scientific Translations, Jerusalem, 1965.
[27] E.P. Dolzhenko, “Boundary value uniqueness theorems for analytic functions,” Math. Notes, 26 (6), pp. 437- 442, 1979.

Spectral Analysis of Non-selfadjoint Second Order Difference Equation with Operator Coefficient

Yıl 2020, Cilt: 24 Sayı: 3, 494 - 500, 01.06.2020

Gökhan Mutlu Esra Kır Arpat

https://doi.org/10.16984/saufenbilder.627496

Cited By: 4

Öz

In this paper, we consider the discrete Sturm-Liouville operator generated by second order difference equation with non-selfadjoint operator coefficient. This operator is the discrete analogue of the Sturm-Liouville differential operator generated by Sturm-Liouville operator equation which has been studied in detail. We find the Jost solution of this operator and examine its asymptotic and analytical properties. Then, we find the continuous spectrum, the point spectrum and the set of spectral singularities of this discrete operator. We finally prove that this operator has a finite number of eigenvalues and spectral singularities under a specific condition.

Anahtar Kelimeler

Sturm-Liouville’s operator equation, Non-selfadjoint operators, Discrete operators, Continuous spectrum, Operator coefficients

Kaynakça

[1] M.A. Naimark, “Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint operator of second order on a semi-axis,” AMS Transl. 2 (16), pp. 103-193, 1960.
[2] M.A. Naimark, “Linear Differential Operators, II,” Ungar, New York, 1968.
[3] V.E. Lyance, “A differential operator with spectral singularities, I-II,” AMS Transl. 2 (60), pp. 185-225, 227-283, 1967.
[4] B.S. Pavlov, “The non-selfadjoint Schrödinger operator in Spectral Theory and Wave Processes, Topics in Mathematical Physics, Vol.1, Birman M.S. (eds), Springer, Boston, MA, pp. 87- 114, 1967.
[5] J.T. Schwartz, “Some non-selfadjoint operators,” Comm. Pure Appl. Math., 13, pp. 609-639, 1960.
[6] C.M. Bender, “Making sense of non- Hermitian Hamiltonians,” Rep. Prog. Phys., 70, pp. 947-1018, 2007.
[7] A. Mostafazadeh, “Pseudo-Hermitian representation of quantum mechanics,” arXiv:0810.5643, 2008.
[8] B.F. Samsonov, “SUSY transformations between digonalizable and nondiagonalizable Hamiltonians,” arXiv:quant-ph/0503075, 2005.
[9] A. Mostafazadeh, “Spectral singularities of complex scattering potentials and infinite reaction and transmission coefficients at real energies,” arXiv:0901.4472, 2009.
[10] R.P. Agarwal, “Difference Equations and Inequalities: Theory, Methods and Applications,” Marcel Dekker, New York, 2000.
[11] R.P. Agarwal and P.J.Y. Wong, “Advanced Topics in Difference Equations,” Kluwer, Dordrecht, 1997.
[12] Y. Aygar and E. Bairamov, “Jost solution and the spectral properties of the matrixvalued difference operators,” Appl. Math. Comput., 218, pp. 9676-9681, 2012.
[13] E. Bairamov, Y. Aygar and S. Cebesoy, “Spectral analysis of a selfadjoint matrix valued discrete operator on the whole axis,” J. Nonlinear Sci. Appl., 9, pp. 4257- 4262, 2016.
[14] R. Carlson, “An inverse problem for the matrix Schrödinger equation,” J. Math. Anal. Appl., 267, pp. 564-575, 2002.
[15] S. Clark, Gesztesy and W. Renger, “Trace formulas and Borg-type theorems for matrix valued Jacobi and Dirac finite difference operators,” J. Differential Equations, 219, pp. 144-182, 2005.
[16] F. Gesztesy, A. Kiselev and K.A. Makarov, “Uniqueness results for matrixvalued Schrödinger, Jacobi and Dirac-type operators,” Math. Nachr., 239, pp. 103- 145, 2002.
[17] E. Bairamov, Ö. Çakar and A.M. Krall, “Non-selfadjoint difference operators and Jacobi matrices with spectral singularities,” Math. Nachr., 229, pp. 5- 14, 2001.
[18] A.M. Krall, E.Bairamov and Ö. Çakar, “Spectral analysis of a non-selfadjoint discrete Schrödinger operators with spectral singularities,” Math. Nachr., 231, pp. 89-104, 2001.
[19] T. Koprubasi and R.N. Mohapatra, “Spectral properties of generalized eigenparameter dependent discrete Sturm- Liouville type equation,” Quaestiones Mathematicae, 40 (4), pp. 491-505, 2017.
[20] M. Adivar and E. Bairamov, “Spectral properties of non-selfadjoint difference operators,” J. Math. Anal. Appl., 261, pp. 461-478, 2001.
[21} M. Adivar and E. Bairamov, “Difference equations of second order with spectral singularities,” J. Math. Anal. Appl., 277, pp. 714-721, 2003.
[22] S. Cebesoy, Y. Aygar and E. Bairamov, “Matrix valued difference equations with spectral singularities,” International Journal of Mathematical, Computational Physical, Electrical and Computer Engineering, 9 (11), pp. 579-582, 2015.
[23] M.V. Keldysh, “On the completeness of the eigenfunctions of some classes of nonselfadjoint linear operators,” Russ. Math. Surv., 26 (4), pp. 15-44, 1971.
[24] V.P. Serebrjakov, “An inverse problem of the scattering theory for difference equations with matrix coeffcients,” Dokl. Akad. Nauk SSSR, 250 (3), pp. 562-565, 1980 (in Russian).
[25] L.A. Lusternik and V.I. Sobolev, “Elements of Functional Analysis,” Halsted Press, New York, 1974.
[26] I.M. Glazman, “Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators,” Israel Program for Scientific Translations, Jerusalem, 1965.
[27] E.P. Dolzhenko, “Boundary value uniqueness theorems for analytic functions,” Math. Notes, 26 (6), pp. 437- 442, 1979.

Toplam 27 adet kaynakça vardır.

Ayrıntılar

Birincil Dil	İngilizce
Konular	Matematik
Bölüm	Araştırma Makalesi
Yazarlar	Gökhan Mutlu 0000-0002-0674-2908 Esra Kır Arpat 0000-0002-6322-5130
Yayımlanma Tarihi	1 Haziran 2020
Gönderilme Tarihi	1 Ekim 2019
Kabul Tarihi	20 Mart 2020
Yayımlandığı Sayı	Yıl 2020 Cilt: 24 Sayı: 3

Kaynak Göster

APA	Mutlu, G., & Kır Arpat, E. (2020). Spectral Analysis of Non-selfadjoint Second Order Difference Equation with Operator Coefficient. Sakarya University Journal of Science, 24(3), 494-500. https://doi.org/10.16984/saufenbilder.627496
AMA	Mutlu G, Kır Arpat E. Spectral Analysis of Non-selfadjoint Second Order Difference Equation with Operator Coefficient. SAUJS. Haziran 2020;24(3):494-500. doi:10.16984/saufenbilder.627496
Chicago	Mutlu, Gökhan, ve Esra Kır Arpat. “Spectral Analysis of Non-Selfadjoint Second Order Difference Equation With Operator Coefficient”. Sakarya University Journal of Science 24, sy. 3 (Haziran 2020): 494-500. https://doi.org/10.16984/saufenbilder.627496.
EndNote	Mutlu G, Kır Arpat E (01 Haziran 2020) Spectral Analysis of Non-selfadjoint Second Order Difference Equation with Operator Coefficient. Sakarya University Journal of Science 24 3 494–500.
IEEE	G. Mutlu ve E. Kır Arpat, “Spectral Analysis of Non-selfadjoint Second Order Difference Equation with Operator Coefficient”, SAUJS, c. 24, sy. 3, ss. 494–500, 2020, doi: 10.16984/saufenbilder.627496.
ISNAD	Mutlu, Gökhan - Kır Arpat, Esra. “Spectral Analysis of Non-Selfadjoint Second Order Difference Equation With Operator Coefficient”. Sakarya University Journal of Science 24/3 (Haziran 2020), 494-500. https://doi.org/10.16984/saufenbilder.627496.
JAMA	Mutlu G, Kır Arpat E. Spectral Analysis of Non-selfadjoint Second Order Difference Equation with Operator Coefficient. SAUJS. 2020;24:494–500.
MLA	Mutlu, Gökhan ve Esra Kır Arpat. “Spectral Analysis of Non-Selfadjoint Second Order Difference Equation With Operator Coefficient”. Sakarya University Journal of Science, c. 24, sy. 3, 2020, ss. 494-00, doi:10.16984/saufenbilder.627496.
Vancouver	Mutlu G, Kır Arpat E. Spectral Analysis of Non-selfadjoint Second Order Difference Equation with Operator Coefficient. SAUJS. 2020;24(3):494-500.