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November  2017, 16(6): 2227-2251. doi: 10.3934/cpaa.2017110

## Essential spectral singularities and the spectral expansion for the Hill operator

Received  February 2017 Revised  May 2017 Published  July 2017

In this paper we investigate the spectral expansion for the one-dimensional Schrodinger operator with a periodic complex-valued potential. For this we consider in detail the spectral singularities and introduce new concepts as essential spectral singularities and singular quasimomenta.

Citation: 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
##### References:
 [1] M. S. P. Eastham, The Spectral Theory of Periodic Differential Operators, New York: Hafner, 1974. [2] M. G. Gasymov, Spectral analysis of a class of second-order nonself-adjoint differential operators, Fankts. Anal. Prilozhen, 14 (1980), 14-19. [3] I. M. Gelfand, Expansion in series of eigenfunctions of an equation with periodic coefficients, Sov. Math. Dokl., 73 (1950), 1117-1120. [4] F. Gesztesy and V. Tkachenko, A criterion for Hill's operators to be spectral operators of scalar type, J. Analyse Math., 107 (2009), 287-353. doi: 10.1007/s11854-009-0012-5. [5] W. Magnus and S. Winkler, Hill's Equation, New York: Inter. Publ. , 1966. [6] V. A. Marchenko, Sturm-Liouville Operators and Applications, Birkhauser Verlag, Basel, 1986. doi: 10.1007/978-3-0348-5485-6. [7] D. C. McGarvey, Differential operators with periodic coefficients in Lp(-∞, ∞), Journal of Mathematical Analysis and Applications, 11 (1965), 564-596. doi: 10.1016/0022-247X(65)90105-8. [8] D. C. McGarvey, Perturbation results for periodic differential operators, Journal of Mathematical Analysis and Applications, 12 (1965), 187-234. doi: 10.1016/0022-247X(65)90033-8. [9] V. P. Mikhailov, On the Riesz bases in L2(0, 1), Sov. Math. Dokl., 25 (1962), 981-984. [10] M. A. Naimark, Linear Differential Operators, George G. Harrap, London, 1967. [11] E. C. Titchmarsh, Eigenfunction Expansion (Part II), Oxford Univ. Press, 1958. [12] V. A. Tkachenko, Spectral analysis of nonself-adjoint Schrodinger operator with a periodic complex potential, Sov. Math. Dokl., 5 (1964), 413-415. [13] A. A. Shkalikov, On the Riesz basis property of the root vectors of ordinary differential operators, Russian Math. Surveys, 34 (1979), 249-250. [14] O. A. Veliev, The one dimensional Schrodinger operator with a periodic complex-valued potential, Sov. Math. Dokl., 250 (1980), 1292-1296. [15] O. A. Veliev, The spectrum and spectral singularities of differential operators with complexvalued periodic coefficients, Differential Cprime Nye Uravneniya, 19 (1983), 1316-1324. [16] O. A. Veliev, The spectral resolution of the nonself-adjoint differential operators with periodic coefficients, Differential Cprime Nye Uravneniya, 22 (1986), 2052-2059. [17] O. A. Veliev and M. Toppamuk Duman, The spectral expansion for a nonself-adjoint Hill operators with a locally integrable potential, J. Math. Anal. Appl., 265 (2002), 76-90. doi: 10.1006/jmaa.2001.7693. [18] O. A. Veliev, Uniform convergence of the spectral expansion for a differential operator with periodic matrix coefficients, Boundary Value Problems, Volume 2008, Article ID 628973, 22 pp. (2008). [19] O. A. Veliev, Asymptotic analysis of non-self-adjoint Hill's operators, Central European Journal of Mathematics, 11 (2013), 2234-2256. doi: 10.2478/s11533-013-0305-x. [20] 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.

show all references

##### References:
 [1] M. S. P. Eastham, The Spectral Theory of Periodic Differential Operators, New York: Hafner, 1974. [2] M. G. Gasymov, Spectral analysis of a class of second-order nonself-adjoint differential operators, Fankts. Anal. Prilozhen, 14 (1980), 14-19. [3] I. M. Gelfand, Expansion in series of eigenfunctions of an equation with periodic coefficients, Sov. Math. Dokl., 73 (1950), 1117-1120. [4] F. Gesztesy and V. Tkachenko, A criterion for Hill's operators to be spectral operators of scalar type, J. Analyse Math., 107 (2009), 287-353. doi: 10.1007/s11854-009-0012-5. [5] W. Magnus and S. Winkler, Hill's Equation, New York: Inter. Publ. , 1966. [6] V. A. Marchenko, Sturm-Liouville Operators and Applications, Birkhauser Verlag, Basel, 1986. doi: 10.1007/978-3-0348-5485-6. [7] D. C. McGarvey, Differential operators with periodic coefficients in Lp(-∞, ∞), Journal of Mathematical Analysis and Applications, 11 (1965), 564-596. doi: 10.1016/0022-247X(65)90105-8. [8] D. C. McGarvey, Perturbation results for periodic differential operators, Journal of Mathematical Analysis and Applications, 12 (1965), 187-234. doi: 10.1016/0022-247X(65)90033-8. [9] V. P. Mikhailov, On the Riesz bases in L2(0, 1), Sov. Math. Dokl., 25 (1962), 981-984. [10] M. A. Naimark, Linear Differential Operators, George G. Harrap, London, 1967. [11] E. C. Titchmarsh, Eigenfunction Expansion (Part II), Oxford Univ. Press, 1958. [12] V. A. Tkachenko, Spectral analysis of nonself-adjoint Schrodinger operator with a periodic complex potential, Sov. Math. Dokl., 5 (1964), 413-415. [13] A. A. Shkalikov, On the Riesz basis property of the root vectors of ordinary differential operators, Russian Math. Surveys, 34 (1979), 249-250. [14] O. A. Veliev, The one dimensional Schrodinger operator with a periodic complex-valued potential, Sov. Math. Dokl., 250 (1980), 1292-1296. [15] O. A. Veliev, The spectrum and spectral singularities of differential operators with complexvalued periodic coefficients, Differential Cprime Nye Uravneniya, 19 (1983), 1316-1324. [16] O. A. Veliev, The spectral resolution of the nonself-adjoint differential operators with periodic coefficients, Differential Cprime Nye Uravneniya, 22 (1986), 2052-2059. [17] O. A. Veliev and M. Toppamuk Duman, The spectral expansion for a nonself-adjoint Hill operators with a locally integrable potential, J. Math. Anal. Appl., 265 (2002), 76-90. doi: 10.1006/jmaa.2001.7693. [18] O. A. Veliev, Uniform convergence of the spectral expansion for a differential operator with periodic matrix coefficients, Boundary Value Problems, Volume 2008, Article ID 628973, 22 pp. (2008). [19] O. A. Veliev, Asymptotic analysis of non-self-adjoint Hill's operators, Central European Journal of Mathematics, 11 (2013), 2234-2256. doi: 10.2478/s11533-013-0305-x. [20] 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.
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