A polynomial-type Jost solution and spectral properties of a self-adjoint quantum-difference operator. (English) Zbl 1354.39012
Summary: In this paper, we find a polynomial-type Jost solution of a self-adjoint \(q\)-difference equation of second order. Then we investigate the analytical properties and asymptotic behavior of the Jost solution. We prove that the self-adjoint operator \(L\) generated by the \(q\)-difference expression of second order has essential spectrum filling the segment \([-2\sqrt{q},2\sqrt{q}]\), \(q>1\). Finally, we examine the properties of the eigenvalues of \(L\).
MSC:
| 39A70 | Difference operators |
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
| 39A12 | Discrete version of topics in analysis |
| 34L05 | General spectral theory of ordinary differential operators |
References:
| [1] | Adıvar, M., Bairamov, E.: Spectral properties of non-selfadjoint difference operators. J. Math. Anal. Appl. 261(2), 461-478 (2001) · Zbl 0992.39018 · doi:10.1006/jmaa.2001.7532 |
| [2] | Adıvar, M., Bohner, M.: Spectral analysis of \[q\] q-difference equations with spectral singularities. Math. Comput. Model. 43(7-8), 695-703 (2006) · Zbl 1134.39018 · doi:10.1016/j.mcm.2005.04.014 |
| [3] | Adıvar, M., Bohner, M.: Spectrum and principal vectors of second order \[q\] q-difference equations. Indian J. Math. 48(1), 17-33 (2006) · Zbl 1114.39003 |
| [4] | Agarwal, R.P.: Difference equations and inequalities. Monographs and Textbooks in Pure and Applied Mathematics, vol. 228. Theory, methods, and applications, 2nd edn. Marcel Dekker, New York (2000) · Zbl 0952.39001 |
| [5] | Aygar, Y., Bairamov, E.: Jost solution and the spectral properties of the matrix-valued difference operators. Appl. Math. Comput. 218(19), 9676-9681 (2012) · Zbl 1246.39003 |
| [6] | Bairamov, E., Aygar, Y., Olgun, M.: Jost solution and the spectrum of the discrete Dirac systems. Bound. Value Probl., art. ID 306571, 11 (2010) · Zbl 1210.39003 |
| [7] | Bairamov, E., Çakar, Ö., Çelebi, A.O.: Quadratic pencil of Schrödinger operators with spectral singularities: discrete spectrum and principal functions. J. Math. Anal. Appl. 216(1), 303-320 (1997) · Zbl 0892.34077 · doi:10.1006/jmaa.1997.5689 |
| [8] | Bairamov, E., Çakar, Ö., Krall, A.M.: An eigenfunction expansion for a quadratic pencil of a Schrödinger operator with spectral singularities. J. Differ. Equ. 151(2), 268-289 (1999) · Zbl 0945.34067 · doi:10.1006/jdeq.1998.3518 |
| [9] | Bairamov, E., Çelebi, A.O.: Spectrum and spectral expansion for the non-selfadjoint discrete Dirac operators. Q. J. Math. Oxford Ser. (2) 50(200), 371-384 (1999) · Zbl 0945.47026 |
| [10] | Bairamov, E., Coskun, C.: The structure of the spectrum of a system of difference equations. Appl. Math. Lett. 18(4), 387-394 (2005) · Zbl 1082.39012 · doi:10.1016/j.aml.2004.01.009 |
| [11] | Bairamov, E., Karaman, Ö.: Spectral singularities of Klein-Gordon \[s\] s-wave equations with an integral boundary condition. Acta Math. Hung. 97(1-2), 121-131 (2002) · Zbl 1126.34379 · doi:10.1023/A:1020815113773 |
| [12] | Bohner, M., Peterson, A.: Dynamic equations on time scales. Birkhäuser, Boston (2001) (an introduction with applications) · Zbl 0978.39001 |
| [13] | Bohner, M., Guseinov, G., Peterson, A.: Introduction to the time scales calculus. In: Advances in Dynamic Equations on Time Scales, pp. 1-15. Birkhäuser, Boston (2003) · Zbl 1114.39003 |
| [14] | Chadan, K., Sabatier, P.C.: Inverse problems in quantum scattering theory. With a foreword by R. G. Newton. Texts and Monographs in Physics. Springer, New York (1977) · Zbl 0363.47006 |
| [15] | Kac, V., Cheung, P.: Quantum calculus. Universitext. Springer, New York (2002) · Zbl 0986.05001 · doi:10.1007/978-1-4613-0071-7 |
| [16] | Levitan, B.M.: Inverse Sturm-Liouville Problems. VSP, Zeist (1987) (translated from the Russian by O. Efimov) · Zbl 0749.34001 |
| [17] | Lusternik, L.A., Sobolev, V.J.: Elements of functional analysis. International Monographs on Advanced Mathematics & Physics. Hindustan, Delhi (1974) (russian edition) · Zbl 0293.46001 |
| [18] | Marchenko, V.A.: Sturm-Liouville operators and applications. Operator Theory: Advances and Applications, vol. 22. Birkhäuser, Basel (1986) (translated from the Russian by A. Iacob) · Zbl 0592.34011 |
| [19] | Naĭmark, M.A.: Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint differential operator of the second order on a semi-axis. Am. Math. Soc. Transl. 2(16), 103-193 (1960) · Zbl 0100.29504 · doi:10.1090/trans2/016/02 |
| [20] | Naĭmark, M.A.: Linear differential operators. Part II: Linear differential operators in Hilbert space. Frederick Ungar, New York (1968) (with additional material by the author, and a supplement by V. È. Ljance. Translated from the Russian by E. R. Dawson. English translation edited by W.N. Everitt) · Zbl 0227.34020 |
| [21] | Weyl, H.: Über beschränkte quadratische Formen, deren Differenz vollständig ist. Rend. Circ. Mat. Palermo 27(1), 373-392 (1909) · JFM 40.0395.01 · doi:10.1007/BF03019655 |
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