# Non-accretive Sturm–Liouville operator with discrete spectrum

### Abstract

For the first time, the Sturm-Liouville equations with a complex potential were studied by M.A. Naimark. M.A. Naimark managed to find sufficient conditions for a complex potential when the corresponding Sturm-Liouville operator on the semi-axis has a discrete spectrum. later, the result of M.A. Naimark was strengthened in the works of V.B. Lidskii. the conditions for the complex potential given by V.B. Lidskii guarantee the accretivity of the studied Sturm-Liouville operators. the question of the existence of non-discrete Sturm-Liouville operators with a discrete spectrum remained relevant. The proposed article provides an answer to this question. For the Sturm - Liouville equation with a complex potential, we have constructed a special solution that decreases at infinity and, for each fixed value of the independent variable, is an entire function of the spectral parameter. Using this solution, a generalization of the well-known theorem of V.B. Lidskii on the conditions on the potential under which the spectrum of the corresponding Sturm – Liouville operator is discrete and the system of root vectors is complete and minimal. In contrast to Lidskii’s work, instead of bounded below the real part or semi-boundedness of the imaginary part of the potential, it is only required that the region of potential values lie outside a certain angle of an arbitrary opening with a bisector along the negative real semiaxis.### References

[1] M.A. Naimark, Investigation of the spectrum and the expansion in eigenfunctions of a nonselfadjoint operator of the second order on a semi-axis (Trudy Moskov. Mat. Obsc., 3 (1954), 181–270) [in Russian].

[2] I.M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators (Israel Program for Scientific Trans., Jerusalem, 1965).

[3] H. Weyl, "Uber gew¨ohnliche Differentialgleichungen mit Singularit¨aten und die zugeh¨origen Entwicklungen willku¨rlicher¨ Funktionen", Мат. Ann. 68 (1910), 220–269.

[4] M.A. Naimark, "On the spectrum of singular nonselfadjoint differential second-order operators", Dokl. Akad. Nauk SSSR 85:1 (1952), 41–44 [in Russian].

[5] V.B. Lidskii, Conditions for completeness of a system of root subspaces for non-selfadjoint operators with discrete spectrum (Tr. Mosk. Mat. Obs., 8 GIFML, Moscow, (1959), 83–120) [in Russian].

[6] V.B. Lidskii, A non-selfadjoint operator of Sturm–Liouville type with a discrete spectrum (Tr. Mosk. Mat. Obs., 9 (1960), 45–79) [in Russian].

[7] T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Berlin–Heidelberg–New York, 1966).

[8] D.B. Sears, "Note on the uniqueness of the Green’s functions associated with certain differential equations", Canadian J. of Math. 2:3(1950), 314–325.

[9] R. Mennicken, M. M¨oller, Non-Self-Adjoint Boundary Eigenvalue Problems (Elsevier, Amsterdam–London, 2003). Zbl 1033.34001

[10] F. Bagarello, J.P. Gazeau, F.H. Szafraniec, M. Znojil, Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects (Hoboken: John Wiley and Sons, New Jersey, 2015).

[11] J. Sj¨ostrand, Spectral instability for non-selfadjoint operators (Palaiseau Cedex, Preprint / Ecole Polytechnique, 2002).

[12] E.B. Davies, "Non-self-adjoint differential operators", Bull. London Math. Soc. 34:5(2002), 513–532.

[13] Kh.K. Ishkin, "On the Spectral Instability of the Sturm–Liouville Operator with a Complex Potential", Differ. Equ. 45:4 (2009), 494–509 (Translated from Differ. Uravneniya 45:4 (2009), 480–495).

[14] Kh.K. Ishkin, "A localization criterion for the eigenvalues of a spectrally unstable operator", Doklady Mathematics 80:3 (2009), 829–832 (Translated from Dokl. AN 429:3 (2009), 301–304).

[15] Kh.K. Ishkin, "Conditions of Spectrum Localization for Operators not Close to Self-Adjoint Operators", Doklady Mathematics 97:2 (2018), 170–173 (Translated from Dokl. AN 479:5 (2018), 497–500).

[16] E.B. Davies, "Eigenvalues of an elliptic system", Math. Zeitschrift 243 (2003), 719–743.

[17] Kh.K. Ishkin, "On localization of the spectrum of the problem with complex weight", Journal of Mathematical Sciences 150:6 (2008), 2488–2499 (Translated from Fundamentalnaya i Prikladnaya Matematika 12:5 (2006), 49–64).

[18] Kh.K. Ishkin, "On analytic properties of Weyl function of Sturm–Liouville operator with a decaying complex potential", Ufa Math. Journal 5:1 (2013), 36–55.

[19] Kh.K. Ishkin, "On a Trivial Monodromy Criterion for the Sturm–Liouville Equation", Math. Notes 94:4 (2013), 508–523 (Translated from Matem. Zametki 94:4 (2013), 552–568).

[20] A.M. Savchuk, A.A. Shkalikov, "Spectral Properties of the Complex Airy Operator on the Half-Line", Funct. Anal. Appl. 51:1 (2017), 66–79 (Translated from Funk. analiz i ego pril. 51:1 (2017), 82–98).

[21] Kh.K. Ishkin, "A localization criterion for the spectrum of the Sturm-Liouville operator on a curve", St. Petersburg Math. J. 28:1 (2017), 37–63 (Translated from Algebra i Analiz 28:1 (2016), 52–88).

[22] Kh.K. Ishkin, A.V. Rezbayev, "Toward the Davies formula on the distribution of the eigenvalues of a nonselfadjoint differential operator", Complex analysis, Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh. Temat. Obz., VINITI 153 (2018), 84–93 [in Russian]. [23] M.V. Fedoryuk, Asymptotic Analysis: Linear Ordinary Differential Equations (Springer-Verlag, Berlin, 1993). Zbl 0782.34001

[23] M.V. Fedoryuk, Asymptotic Analysis: Linear Ordinary Differential Equations (Springer-Verlag, Berlin, 1993). Zbl 0782.34001

[24] B.Ya. Levin, Distribution of Zeros of Entire Functions (Gostekhizdat, Moscow, 1956); English transl., Amer. Math. Soc.,

Providence, RI, 1964.

[2] I.M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators (Israel Program for Scientific Trans., Jerusalem, 1965).

[3] H. Weyl, "Uber gew¨ohnliche Differentialgleichungen mit Singularit¨aten und die zugeh¨origen Entwicklungen willku¨rlicher¨ Funktionen", Мат. Ann. 68 (1910), 220–269.

[4] M.A. Naimark, "On the spectrum of singular nonselfadjoint differential second-order operators", Dokl. Akad. Nauk SSSR 85:1 (1952), 41–44 [in Russian].

[5] V.B. Lidskii, Conditions for completeness of a system of root subspaces for non-selfadjoint operators with discrete spectrum (Tr. Mosk. Mat. Obs., 8 GIFML, Moscow, (1959), 83–120) [in Russian].

[6] V.B. Lidskii, A non-selfadjoint operator of Sturm–Liouville type with a discrete spectrum (Tr. Mosk. Mat. Obs., 9 (1960), 45–79) [in Russian].

[7] T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Berlin–Heidelberg–New York, 1966).

[8] D.B. Sears, "Note on the uniqueness of the Green’s functions associated with certain differential equations", Canadian J. of Math. 2:3(1950), 314–325.

[9] R. Mennicken, M. M¨oller, Non-Self-Adjoint Boundary Eigenvalue Problems (Elsevier, Amsterdam–London, 2003). Zbl 1033.34001

[10] F. Bagarello, J.P. Gazeau, F.H. Szafraniec, M. Znojil, Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects (Hoboken: John Wiley and Sons, New Jersey, 2015).

[11] J. Sj¨ostrand, Spectral instability for non-selfadjoint operators (Palaiseau Cedex, Preprint / Ecole Polytechnique, 2002).

[12] E.B. Davies, "Non-self-adjoint differential operators", Bull. London Math. Soc. 34:5(2002), 513–532.

[13] Kh.K. Ishkin, "On the Spectral Instability of the Sturm–Liouville Operator with a Complex Potential", Differ. Equ. 45:4 (2009), 494–509 (Translated from Differ. Uravneniya 45:4 (2009), 480–495).

[14] Kh.K. Ishkin, "A localization criterion for the eigenvalues of a spectrally unstable operator", Doklady Mathematics 80:3 (2009), 829–832 (Translated from Dokl. AN 429:3 (2009), 301–304).

[15] Kh.K. Ishkin, "Conditions of Spectrum Localization for Operators not Close to Self-Adjoint Operators", Doklady Mathematics 97:2 (2018), 170–173 (Translated from Dokl. AN 479:5 (2018), 497–500).

[16] E.B. Davies, "Eigenvalues of an elliptic system", Math. Zeitschrift 243 (2003), 719–743.

[17] Kh.K. Ishkin, "On localization of the spectrum of the problem with complex weight", Journal of Mathematical Sciences 150:6 (2008), 2488–2499 (Translated from Fundamentalnaya i Prikladnaya Matematika 12:5 (2006), 49–64).

[18] Kh.K. Ishkin, "On analytic properties of Weyl function of Sturm–Liouville operator with a decaying complex potential", Ufa Math. Journal 5:1 (2013), 36–55.

[19] Kh.K. Ishkin, "On a Trivial Monodromy Criterion for the Sturm–Liouville Equation", Math. Notes 94:4 (2013), 508–523 (Translated from Matem. Zametki 94:4 (2013), 552–568).

[20] A.M. Savchuk, A.A. Shkalikov, "Spectral Properties of the Complex Airy Operator on the Half-Line", Funct. Anal. Appl. 51:1 (2017), 66–79 (Translated from Funk. analiz i ego pril. 51:1 (2017), 82–98).

[21] Kh.K. Ishkin, "A localization criterion for the spectrum of the Sturm-Liouville operator on a curve", St. Petersburg Math. J. 28:1 (2017), 37–63 (Translated from Algebra i Analiz 28:1 (2016), 52–88).

[22] Kh.K. Ishkin, A.V. Rezbayev, "Toward the Davies formula on the distribution of the eigenvalues of a nonselfadjoint differential operator", Complex analysis, Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh. Temat. Obz., VINITI 153 (2018), 84–93 [in Russian]. [23] M.V. Fedoryuk, Asymptotic Analysis: Linear Ordinary Differential Equations (Springer-Verlag, Berlin, 1993). Zbl 0782.34001

[23] M.V. Fedoryuk, Asymptotic Analysis: Linear Ordinary Differential Equations (Springer-Verlag, Berlin, 1993). Zbl 0782.34001

[24] B.Ya. Levin, Distribution of Zeros of Entire Functions (Gostekhizdat, Moscow, 1956); English transl., Amer. Math. Soc.,

Providence, RI, 1964.

How to Cite

ISHKIN, Kh. K.; MARVANOV, R. I..
Non-accretive Sturm–Liouville operator with discrete spectrum.

**Journal of Mathematics, Mechanics and Computer Science**, [S.l.], v. 108, n. 4, p. 18-25, dec. 2020. ISSN 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/816>. Date accessed: 20 jan. 2021. doi: https://doi.org/10.26577/JMMCS.2020.v108.i4.02.
Section

Mathematics

Keywords
spectral instability, spectrum localization, Sturm–Liouville equation, trivial monodromy