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March  2020, 19(3): 1537-1562. doi: 10.3934/cpaa.2020077

On the spectrality and spectral expansion of the non-self-adjoint mathieu-hill operator in $ L_{2}(-\infty, \infty) $

Dogus University, Acıbadem, Kadiköy, Istanbul, Turkey

Received  June 2019 Revised  September 2019 Published  November 2019

In this paper we investigate the non-self-adjoint operator$ \ H $ generated in $ L_{2}(-\infty, \infty) $ by the Mathieu-Hill equation with a complex-valued potential. We find a necessary and sufficient conditions on the potential for which $ H $ has no spectral singularity at infinity and it is an asymptotically spectral operator. Moreover, we give a detailed classification, stated in term of the potential, for the form of the spectral decomposition of the operator $ H $ by investigating the essential spectral singularities.

Citation: O. A. Veliev. On the spectrality and spectral expansion of the non-self-adjoint mathieu-hill operator in $ L_{2}(-\infty, \infty) $. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1537-1562. doi: 10.3934/cpaa.2020077
References:
[1]

L. V. Ahlfors, Complex Analysis, McGRAW-HILL, 1979.,  Google Scholar

[2]

P. Djakov and B. S. Mitjagin, Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials, Mathematische Annalen, 351 (2011), 509-540.  doi: 10.1007/s00208-010-0612-5.  Google Scholar

[3]

M. S. P. Eastham, The Spectral Theory of Periodic Differential Operators, New York: Hafner, 1974.  Google Scholar

[4]

M. G. Gasymov, Spectral analysis of a class of second-order nonself-adjoint differential operators, Fankts. Anal. Prilozhen, 14 (1980), 14-19.   Google Scholar

[5]

I. M. Gelfand, Expansion in series of eigenfunctions of an equation with periodic coefficients, Sov. Math. Dokl., 73 (1950), 1117-1120.   Google Scholar

[6]

F. Gesztesy and V. Tkachenko, A criterion for Hill operators to be spectral operators of scalar type, J. Analyse Math., 107 (2009), 287–353. doi: 10.1007/s11854-009-0012-5.  Google Scholar

[7]

N. B. Kerimov, On a Boundary value problem of N. I. Ionkin type, Differential Equations, 49 (2013), 1233–1245. doi: 10.1134/S0012266113100042.  Google Scholar

[8]

D. McGarvey, Operators commuting with translations by one. Part Ⅱ. Differential operators with periodic coefficients in $L_{p}(-\infty, \infty)$, J. Math. Anal. Appl., 11 (1965), 564–596. doi: 10.1016/0022-247X(65)90105-8.  Google Scholar

[9]

D. McGarvey, Operators commuting with translations by one. Part Ⅲ. Perturbation results for periodic differential operators, J. Math. Anal. Appl., 12 (1965), 187–234. doi: 10.1016/0022-247X(65)90033-8.  Google Scholar

[10]

M. A. Naimark, Linear Differential Operators, George G. Harrap, London, 1967.  Google Scholar

[11]

A. A. Shkalikov and O. A. Veliev, On the Riesz basis property of the eigen- and associated functions of periodic and antiperiodic Sturm-Liouville problems, Math. Notes, 85 (2009), 647–660. doi: 10.1134/S0001434609050058.  Google Scholar

[12]

O. A. Veliev and M. Toppamuk Duman, The spectral expansion for a non-self-adjoint Hill operators with a locally integrable potential, J. Math. Anal. Appl., 265 (2002), 76–90. doi: 10.1006/jmaa.2001.7693.  Google Scholar

[13]

O. A. Veliev, Asymptotic Analysis of Non-self-adjoint Hill Operators, Cent. Eur. J. Math., 11 (2013), 2234–2256. doi: 10.2478/s11533-013-0305-x.  Google Scholar

[14]

O. A. Veliev, On the simplicity of the eigenvalues of the non-self-adjoint Mathieu-Hill operators, Applied and Computational Mathematics, 13 (2014), 122-134.   Google Scholar

[15]

O. A. Veliev, Spectral problems of a class of non-self-adjoint one-dimensional Schrodinger operators, Journal of Mathematical Analysis and Applications, 422 (2015), 1390–1401. doi: 10.1016/j.jmaa.2014.09.074.  Google Scholar

[16]

O. A. Veliev, On the spectral singularities and spectrality of the Hill's Operator, Operators and Matrices, 10 (2016), 57–71. doi: 10.7153/oam-10-05.  Google Scholar

[17]

O. A. Veliev, Essential spectral singularities and the spectral expansion for the Hill operator, Communication on Pure and Applied Analysis, 16 (2017), 2227–2251. doi: 10.3934/cpaa.2017110.  Google Scholar

[18]

O. A. Veliev, Spectral expansion series with parenthesis for the non-self-adjoint periodic differential operators, Communication on Pure and Applied Analysis, 18 (2019), 397–424. doi: 10.3934/cpaa.2019020.  Google Scholar

show all references

References:
[1]

L. V. Ahlfors, Complex Analysis, McGRAW-HILL, 1979.,  Google Scholar

[2]

P. Djakov and B. S. Mitjagin, Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials, Mathematische Annalen, 351 (2011), 509-540.  doi: 10.1007/s00208-010-0612-5.  Google Scholar

[3]

M. S. P. Eastham, The Spectral Theory of Periodic Differential Operators, New York: Hafner, 1974.  Google Scholar

[4]

M. G. Gasymov, Spectral analysis of a class of second-order nonself-adjoint differential operators, Fankts. Anal. Prilozhen, 14 (1980), 14-19.   Google Scholar

[5]

I. M. Gelfand, Expansion in series of eigenfunctions of an equation with periodic coefficients, Sov. Math. Dokl., 73 (1950), 1117-1120.   Google Scholar

[6]

F. Gesztesy and V. Tkachenko, A criterion for Hill operators to be spectral operators of scalar type, J. Analyse Math., 107 (2009), 287–353. doi: 10.1007/s11854-009-0012-5.  Google Scholar

[7]

N. B. Kerimov, On a Boundary value problem of N. I. Ionkin type, Differential Equations, 49 (2013), 1233–1245. doi: 10.1134/S0012266113100042.  Google Scholar

[8]

D. McGarvey, Operators commuting with translations by one. Part Ⅱ. Differential operators with periodic coefficients in $L_{p}(-\infty, \infty)$, J. Math. Anal. Appl., 11 (1965), 564–596. doi: 10.1016/0022-247X(65)90105-8.  Google Scholar

[9]

D. McGarvey, Operators commuting with translations by one. Part Ⅲ. Perturbation results for periodic differential operators, J. Math. Anal. Appl., 12 (1965), 187–234. doi: 10.1016/0022-247X(65)90033-8.  Google Scholar

[10]

M. A. Naimark, Linear Differential Operators, George G. Harrap, London, 1967.  Google Scholar

[11]

A. A. Shkalikov and O. A. Veliev, On the Riesz basis property of the eigen- and associated functions of periodic and antiperiodic Sturm-Liouville problems, Math. Notes, 85 (2009), 647–660. doi: 10.1134/S0001434609050058.  Google Scholar

[12]

O. A. Veliev and M. Toppamuk Duman, The spectral expansion for a non-self-adjoint Hill operators with a locally integrable potential, J. Math. Anal. Appl., 265 (2002), 76–90. doi: 10.1006/jmaa.2001.7693.  Google Scholar

[13]

O. A. Veliev, Asymptotic Analysis of Non-self-adjoint Hill Operators, Cent. Eur. J. Math., 11 (2013), 2234–2256. doi: 10.2478/s11533-013-0305-x.  Google Scholar

[14]

O. A. Veliev, On the simplicity of the eigenvalues of the non-self-adjoint Mathieu-Hill operators, Applied and Computational Mathematics, 13 (2014), 122-134.   Google Scholar

[15]

O. A. Veliev, Spectral problems of a class of non-self-adjoint one-dimensional Schrodinger operators, Journal of Mathematical Analysis and Applications, 422 (2015), 1390–1401. doi: 10.1016/j.jmaa.2014.09.074.  Google Scholar

[16]

O. A. Veliev, On the spectral singularities and spectrality of the Hill's Operator, Operators and Matrices, 10 (2016), 57–71. doi: 10.7153/oam-10-05.  Google Scholar

[17]

O. A. Veliev, Essential spectral singularities and the spectral expansion for the Hill operator, Communication on Pure and Applied Analysis, 16 (2017), 2227–2251. doi: 10.3934/cpaa.2017110.  Google Scholar

[18]

O. A. Veliev, Spectral expansion series with parenthesis for the non-self-adjoint periodic differential operators, Communication on Pure and Applied Analysis, 18 (2019), 397–424. doi: 10.3934/cpaa.2019020.  Google Scholar

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