American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Nontrivial solutions for the choquard equation with indefinite linear part and upper critical exponent
  • CPAA Home
  • This Issue
  • Next Article
    Unique strong solutions and V-attractor of a three dimensional globally modified magnetohydrodynamic equations
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) $

O. A. Veliev

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.

Keywords: Mathieu-Hill operator, spectral operator, spectral expansion.
Mathematics Subject Classification: 34L05, 34L20.
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

[1]

O. A. Veliev. Essential spectral singularities and the spectral expansion for the Hill operator. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2227-2251. doi: 10.3934/cpaa.2017110
[2]

Eduardo Lara, Rodolfo Rodríguez, Pablo Venegas. Spectral approximation of the curl operator in multiply connected domains. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 235-253. doi: 10.3934/dcdss.2016.9.235
[3]

Mark F. Demers, Hong-Kun Zhang. Spectral analysis of the transfer operator for the Lorentz gas. Journal of Modern Dynamics, 2011, 5 (4) : 665-709. doi: 10.3934/jmd.2011.5.665
[4]

Mario Ahues, Filomena D. d'Almeida, Alain Largillier, Paulo B. Vasconcelos. Defect correction for spectral computations for a singular integral operator. Communications on Pure & Applied Analysis, 2006, 5 (2) : 241-250. doi: 10.3934/cpaa.2006.5.241
[5]

Laurent Amour, Jérémy Faupin. Inverse spectral results in Sobolev spaces for the AKNS operator with partial informations on the potentials. Inverse Problems & Imaging, 2013, 7 (4) : 1115-1122. doi: 10.3934/ipi.2013.7.1115
[6]

Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247
[7]

Grégoire Allaire, Yves Capdeboscq, Marjolaine Puel. Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 1-31. doi: 10.3934/dcdsb.2012.17.1
[8]

Sébastien Gadat, Laurent Miclo. Spectral decompositions and $\mathbb{L}^2$-operator norms of toy hypocoercive semi-groups. Kinetic & Related Models, 2013, 6 (2) : 317-372. doi: 10.3934/krm.2013.6.317
[9]

Adina Juratoni, Flavius Pater, Olivia Bundău. Operator representations of logmodular algebras which admit $\gamma-$spectral $\rho-$dilations. Electronic Research Announcements, 2012, 19: 49-57. doi: 10.3934/era.2012.19.49
[10]

Rúben Sousa, Semyon Yakubovich. The spectral expansion approach to index transforms and connections with the theory of diffusion processes. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2351-2378. doi: 10.3934/cpaa.2018112
[11]

Oktay Veliev. Spectral expansion series with parenthesis for the nonself-adjoint periodic differential operators. Communications on Pure & Applied Analysis, 2019, 18 (1) : 397-424. doi: 10.3934/cpaa.2019020
[12]

Benoît Pausader, Walter A. Strauss. Analyticity of the nonlinear scattering operator. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 617-626. doi: 10.3934/dcds.2009.25.617
[13]

Vittorio Martino. On the characteristic curvature operator. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1911-1922. doi: 10.3934/cpaa.2012.11.1911
[14]

Alexandre I. Danilenko, Mariusz Lemańczyk. Spectral multiplicities for ergodic flows. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4271-4289. doi: 10.3934/dcds.2013.33.4271
[15]

Virginie Bonnaillie-Noël, Corentin Léna. Spectral minimal partitions of a sector. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 27-53. doi: 10.3934/dcdsb.2014.19.27
[16]

Michael Baake, Daniel Lenz. Spectral notions of aperiodic order. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 161-190. doi: 10.3934/dcdss.2017009
[17]

Peter C. Gibson. On the measurement operator for scattering in layered media. Inverse Problems & Imaging, 2017, 11 (1) : 87-97. doi: 10.3934/ipi.2017005
[18]

Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040
[19]

Yunmei Chen, Xianqi Li, Yuyuan Ouyang, Eduardo Pasiliao. Accelerated bregman operator splitting with backtracking. Inverse Problems & Imaging, 2017, 11 (6) : 1047-1070. doi: 10.3934/ipi.2017048
[20]

Dieter Mayer, Tobias Mühlenbruch, Fredrik Strömberg. The transfer operator for the Hecke triangle groups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2453-2484. doi: 10.3934/dcds.2012.32.2453

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (10)
  • HTML views (17)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences