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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

Depart. of Math., Dogus University, Acıbadem, 34722, Kadiköy, Istanbul, Turkey

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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[5]

W. Magnus and S. Winkler, Hill's Equation, New York: Inter. Publ. , 1966.  Google Scholar

[6]

V. A. Marchenko, Sturm-Liouville Operators and Applications, Birkhauser Verlag, Basel, 1986. doi: 10.1007/978-3-0348-5485-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

V. P. Mikhailov, On the Riesz bases in L2(0, 1), Sov. Math. Dokl., 25 (1962), 981-984.   Google Scholar

[10]

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

[11]

E. C. Titchmarsh, Eigenfunction Expansion (Part II), Oxford Univ. Press, 1958. Google Scholar

[12]

V. A. Tkachenko, Spectral analysis of nonself-adjoint Schrodinger operator with a periodic complex potential, Sov. Math. Dokl., 5 (1964), 413-415.   Google Scholar

[13]

A. A. Shkalikov, On the Riesz basis property of the root vectors of ordinary differential operators, Russian Math. Surveys, 34 (1979), 249-250.   Google Scholar

[14]

O. A. Veliev, The one dimensional Schrodinger operator with a periodic complex-valued potential, Sov. Math. Dokl., 250 (1980), 1292-1296.   Google Scholar

[15]

O. A. Veliev, The spectrum and spectral singularities of differential operators with complexvalued periodic coefficients, Differential Cprime Nye Uravneniya, 19 (1983), 1316-1324.   Google Scholar

[16]

O. A. Veliev, The spectral resolution of the nonself-adjoint differential operators with periodic coefficients, Differential Cprime Nye Uravneniya, 22 (1986), 2052-2059.   Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[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.  Google Scholar

[5]

W. Magnus and S. Winkler, Hill's Equation, New York: Inter. Publ. , 1966.  Google Scholar

[6]

V. A. Marchenko, Sturm-Liouville Operators and Applications, Birkhauser Verlag, Basel, 1986. doi: 10.1007/978-3-0348-5485-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

V. P. Mikhailov, On the Riesz bases in L2(0, 1), Sov. Math. Dokl., 25 (1962), 981-984.   Google Scholar

[10]

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

[11]

E. C. Titchmarsh, Eigenfunction Expansion (Part II), Oxford Univ. Press, 1958. Google Scholar

[12]

V. A. Tkachenko, Spectral analysis of nonself-adjoint Schrodinger operator with a periodic complex potential, Sov. Math. Dokl., 5 (1964), 413-415.   Google Scholar

[13]

A. A. Shkalikov, On the Riesz basis property of the root vectors of ordinary differential operators, Russian Math. Surveys, 34 (1979), 249-250.   Google Scholar

[14]

O. A. Veliev, The one dimensional Schrodinger operator with a periodic complex-valued potential, Sov. Math. Dokl., 250 (1980), 1292-1296.   Google Scholar

[15]

O. A. Veliev, The spectrum and spectral singularities of differential operators with complexvalued periodic coefficients, Differential Cprime Nye Uravneniya, 19 (1983), 1316-1324.   Google Scholar

[16]

O. A. Veliev, The spectral resolution of the nonself-adjoint differential operators with periodic coefficients, Differential Cprime Nye Uravneniya, 22 (1986), 2052-2059.   Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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