Spectral properties of non-selfadjoint Sturm-Liouville operator equation on the real axis

Abstract

 In this paper, we analyze the non-selfadjoint Sturm-Liouville operator $L$ defined in the Hilbert space $L_{2}(\mathbb{R},H)$ of vector-valued functions which are strongly-measurable and square-integrable in $ \mathbb{R} $. $L$ is defined

\[L(y)=-y''+Q(x)y,\, x\in\mathbb{R} \]

for every $ y \in L_{2}(\mathbb{R},H) $ where the potential $Q(x)$ is a non-selfadjoint, completely continuous operator in a separable Hilbert space $H$ for each $x\in \mathbb{R}.$ We obtain the Jost solutions of this operator and examine the analytic and asymptotic properties. Moreover, we find the point spectrum and the spectral singularities of $ L $ and also obtain the sufficient condition which assures the finiteness of the eigenvalues and spectral singularities of $ L $.

Keywords

• Sturm-Liouville operator equation
• eigenvalues
• spectral singularities
• operator coefficient
• non-selfadjoint operators

References

1. [1] Z.S. Agranovic, V.A. Marchenko, The Inverse Problem of Scattering Theory, Gordon and Breach, 1965.
2. [2] E.K. Arpat and G. Mutlu, Spectral properties of Sturm-Liouville system with eigenvalue-dependent boundary conditions, Internat. J. Math. 26 (10), 1550080- 1550088, 2015.
3. [3] E. Bairamov, E.K. Arpat and G. Mutlu, Spectral properties of non-selfadjoint Sturm- Liouville operator with operator coefficient, J. Math. Anal. Appl. 456 (1), 293-306, 2017.
4. [4] E. Bairamov and Ş. Cebesoy, Spectral singularities of the matrix Schrödinger equations, Hacet. J. Math. Stat. 45 (4), 1007-1014, 2016.
5. [5] E. Bairamov and E. Kir, Principal functions of the non-selfadjoint operator generated by system of differential equations, Math. Balkanica (N.S.) 13 (1-2), 85–98, 1999.
6. [6] E. Bairamov and E. Kir, Spectral properties of a finite system of Sturm-Liouville differential operators, Indian J. Pure Appl. Math. 35 (2), 249–256, 2004.
7. [7] E. Bairamov and G.B. Tunca, Discrete spectrum and principial functions of nonselfadjoint differential operator, Czechoslavak Math. J. 49 (124), 689-700, 1999.
8. [8] B.B. Blashak, On the second-order differential operator on the whole axis with spectral singularities (In Russian), Dokl. Akad. Nauk Ukr. SSR I, 38-41, 1966.

9. [9] R. Carlson, An inverse problem for the matrix Schrödinger equation, J. Math. Anal. Appl. 267, 564-575, 2002.
10. [10] S. Clark, F. Gesztesy and W. Renger, Trace formulas and Borg-type theorems for matrix-valued Jacobi and Dirac finite difference operators, J. Differ. Equations 219, 144-182, 2005.
11. [11] C. Coskun and M. Olgun, Principal functions of non-selfadjoint matrix Sturm- Liouville equations, J. Comput. Appl. Math. 235, 4834-4838, 2011.
12. [12] E.P Dolzhenko, Boundary value uniqueness theorems for analytic functions, Math. Notes 26 (6), 437-442, 1979.
13. [13] M.G. Gasymov, Expansion in solutions of the scattering problem for a nonselfadjoint Schrödinger equation (In Russian), Dokl. Akad. Nauk AzSSR 22 (10), 9-12, 1966.
14. [14] M.G. Gasymov, V.V. Zikov and B.M. Levitan, Conditions for discreteness and finiteness of the negative spectrum of Schrödinger’s operator equation (in Russian), Mat. Zametki 2, 531–538, 1967.
15. [15] F. Gesztesy, A. Kiselev and K.A. Makarov, Uniqueness results for matrix-valued Schrodinger, Jacobi and Dirac-type operators, Math. Nachr. 239, 103-145, 2002.
16. [16] M.V. Keldysh, On eigenvalues and eigenfunctions of some classes of nonselfadjoint equations (in Russian), Dokl. Akad. Nauk. SSSR 77 (1), 11-14, 1951.
17. [17] M.V. Keldysh, On the completeness of the eigenfunctions of some classes of nonselfadjoint linear operators, Russ. Math. Surv. 26 (4), 1544, 1971.
18. [18] A.N. Kolmogorov and S.V. Fomin, Introductory Real Analysis, Dover Publ., New York, 1975.
19. [19] A.G. Kostjucenko and B. M. Levitan, Asymptotic behavior of eigenvalues of the operator Sturm-Liouville problem (in Russian), Funkcional. Anal. i Prilozen 1, 86–96, 1967.
20. [20] B.M. Levitan, Investigation of the Green’s function of a Sturm-Liouville equation with an operator coefficient (in Russian), Mat. Sb. (N.S.) 76 (118), 239–270, 1968.
21. [21] B.M. Levitan and G. A. Suvorcenkova, Sufficient conditions for discreteness of the spectrum of a Sturm-Liouville equation with operator coefficient (in Russian), Funkcional. Anal. i Prilozen 2 (2), 56–62, 1968.
22. [22] V.E. Lyance, A differential operator with spectral singularities, II, Amer. Math. Soc. Transl. Ser. 2 60, 227283, 1967.
23. [23] B. Nagy, Operators with spectral singularities, J. Operat. Theor. 15 (2), 307-325, 1986.
24. [24] M.A. Naimark, Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint operator of second order on a semi-axis, Tr. Mosk. Mat. Obs. 3, 181270, 1954.
25. [25] M.A. Naimark, Linear differential operators, II, Ungar, New York, 1968.
26. [26] M. Olgun, Non-selfadjoint matrix Sturm-Liouville operators with eigenvaluedependent boundary conditions, Hacet. J. Math. Stat. 44 (3), 607614, 2015.
27. [27] M. Olgun and C. Coskun, Non-selfadjoint matrix Sturm-Liouville operators with spectral singularities, Appl. Math. Comput. 216, 2271-2275, 2010.
28. [28] B.S. Pavlov, The Nonself-Adjoint Schrödinger Operator, in: Birman M.S. (eds) Spectral Theory andWave Processes, Topics in Mathematical Physics, 1, 87-114, Springer, Boston, MA, 1967.
29. [29] K. Yosida, Functional Analysis, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1980.