Article Contents
Article Contents

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

• 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.

Mathematics Subject Classification: 47E05, 34L05.

 Citation:

•  [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.