Equivalence of Sturm-Liouville Problem with Finitely Many δ-Interactions and Matrix Eigenvalue Problems

Main Article Content

Abdullah Kablan, Mehmet Akif Çetin

Abstract

The purpuse of this article is to show the matrix representations of Sturm-Liouville operators with finitely many δ-interactions. We show that a Sturm-Liouville problem with finitely many δ-interactions can be represented as a finite dimensional matrix eigenvalue problem which has the same eigenvalue with the former Sturm-Liouville operator. Moreover an example is also presented.

Article Details

References

  1. F.V. Atkinson, Discrete and Continuous Boundary Problems, Academic Press-New York, London, 1964.
  2. Q. Kong, H. Wu and A. Zettl, Sturm-Liouville Problems with Finite Spectrum, J. Math. Anal. Appl. 263 (2001), 748-762.
  3. A. Zettl, Sturm-Liouville Theory, Amer. Math. Soc., Math. Surveys Monographs, no. 121, 2005.
  4. J.J. Ao, J. Sun and M.Z. Zhang, The Finite Spectrum of Sturm-Liouville Problems with Transmission Conditions, Appl. Math. Comput. 218 (2011), 1166-1173.
  5. J.J. Ao, J. Sun and M.Z. Zhang, The Finite Spectrum of Sturm-Liouville Problems with Transmission Conditions and Eigenparameter-Dependent Boundary Conditions, Result. Math. 63(3-4) (2013), 1057-1070.
  6. J.J. Ao, F.Z. Bo and J. Sun, Fourth-Order Boundary Value Problems with Finite Spectrum, Appl. Math. Comput. 244 (2014), 952-958.
  7. F.Z. Bo and J.J. Ao, The Finite Spectrum of Fourth-Order Boundary Value Problems with Transmission Conditions, Abstr. Appl. Anal. 2014 (2014), Art. ID 175489.
  8. Q. Kong, H. Volkmer and A. Zettl, Matrix Representations of Sturm-Liouville Problems with Finite Spectrum, Result. Math. 54 (2009), 103-116.
  9. J.J. Ao, J. Sun and M.Z. Zhang, Matrix Representations of Sturm-Liouville Problems with Transmission Conditions, Comput. Math. Appl. 63 (2012), 1335-1348.
  10. J.J. Ao. and J. Sun, Matrix Representations of Sturm-Liouville Problems with Eigenparameter-Dependent Boundary Conditions, Linear Algebra Appl. 438 (2013), 2359-2365.
  11. J.J. Ao. and J. Sun, Matrix Representations of Sturm-Liouville Problems with Coupled Eigenparameter-Dependent Bound- ary Conditions, Appl. Math. Comput. 244 (2014), 142-148.
  12. J.J. Ao, J. Sun and A. Zettl, Matrix Representations of Fourth-Order Boundary Value Problems with Finite Spectrum, Linear Algebra Appl. 436 (2012), 2359-2365.
  13. J.J. Ao, J. Sun and A. Zettl, Equivalance of Fourth-Order Boundary Value Problems and Matrix Eigenvalue Problems, Result. Math. 63 (2013), 581-595.
  14. S. Ge, W. Wang and J.J. Ao, Matrix Representations of Fourth-Order Boundary Value Problems with Periodic Boundary Conditions, Appl. Math. Comput. 227 (2014), 601-609.
  15. A. Kablan and M. D. Manafov, Matrix Representations of Fourth-Order Boundary Value Problems with Transmission Conditions, Mediterranean J. Math. 13(1) (2016), 205-215.
  16. J.J. Ao. and J. Sun, Matrix Representations of Fourth-Order Boundary Value Problems with Coupled or Mixed Boundary Conditions, Linear Multilinear Algebra 63(8) (2015), 1590-1598.
  17. L.N. Pandey and T.F. George, Intersubband Transitions in Quantum Well Heterostructures with Delta-Doped Barriers, Appl. Phys. Lett. 61 (2016), 1081.
  18. R.K. Willardson and A.C. Beer, Semiconductors and Semimetals, Academic Press, London, 1984.
  19. S. Albeverio, F. Gesztesy, R. Høegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, Springer-Verlag, Berlin/New York, 1988.
  20. A. Kablan, M.A. C ¸etin and M.D. Manafov, The Finite Spectrum of Sturm-Liouville Operator with Finitely Many δ- interactions, pre-print.