SPECTRAL PROPERTIES OF SCHRODINGER OPERATOR WITH A GENERAL BOUNDARY CONDITIONS ON FINITE TIME SCALE

Year 2016, Volume: 29 Issue: 2, 467 - 472, 21.06.2016

Esra Kır Arpat , Hatice Terzi

Abstract

In this paper we consider the operator L generated in 𝐿∇2 [𝑎, 𝑏] by the boundary problem−[𝑦∆(𝑡)]∇ + [𝜆 + 𝑞(𝑡)]2𝑦(𝑡) = 0, 𝑡 ∈ [𝑎, 𝑏],𝑦(𝑎) − 𝑘𝑦∆(𝑎) = 0, 𝑦(𝑏) + 𝐾𝑦∆(𝑏) = 0 where 𝑞(𝑡) is partial continuous, 𝑞(𝑡) ≥ 0, 𝑘 ≥ 0,𝐾 ≥ 0. In this paper, spectral properties of Schrodinger problem on finite time scale is examined and the formula of convergent expansion is obtained which is form of series in terms of the eigenfunctions in 𝐿∇2 [𝑎, 𝑏] space.

Keywords

Time scale, delta and nabla derivatives, Schrödinger operator, eigenvalue, eigenfunction.

References

• References
• Huseynov, A. and Bairamov, E., “On Expansions in Eigenfunctions for Second Order Dynamic Equations on Time Scales”, Nonlinear Dynamics and System Theory, 9(1):77-88, (2009).
• Agarwal, R.P., Bohner, M. and Wong, P.J.Y., “Sturm-Liouville Eigenvalue Problems on Time Scales”, Appl. Math. Comput., 99:153-166, (1999).
• Anderson, D.R., Guseinov, G.Sh. and Hoffacker, J., “Higher-order Self-adjoint Boundary Value Problems on Time Scales”, J. Comput. Appl. Math., 194:309-342, (2006).
• Atici, F.M. and Guseinov, G.Sh., “On Green’s Functions and Positive Solutions for Boundary Value Problems on Time Scales”, J. Comput. Appl. Math., 141:75-99, (2002).
• Bohner, M. and Peterson, A., “Dynamics Equations on Time Scales: An Introduction with Applications”, Birkhauser, Boston, (2001).
• Bohner, M. and Peterson, A., “Advances in Dynamics Equations on Time Scales”, Birkhauser, Boston, (2003).
• Chyan, C.J., Davis, J.M., Henderson, J. and Yin, W.K.C., “Eigenvalue Comparisons for Differential Equations on a Measure Chain Electron”, J. Differential Equations, 7pp, (1998).
• Guseinov, G.Sh., “Eigenfunction Expansions for a Sturm-Liouville Problem on Time Scales”, International Journal of Difference Equations, 2:93-104, (2007).