Abstract
We consider the Sturm–Liouville operator T0 on the semi-axis (0,+∞) with the potential eiθq, where 0 < θ < π and q is a real-valued function that may have arbitrarily slow growth at infinity. This operator does not meet any condition of the Keldysh theorem: T0 is non-self-adjoint and its resolvent does not belong to the Neumann–Schatten class 
for any p < ∞. We find conditions for q and perturbations of V under which the localization or the asymptotics of its spectrum is preserved.
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References
J. Aguilar and J. M. Combes, “A class of analytic pertubations for one-body Schrödinger Hamiltonians,” Commun. Math. Phys., 22, 268–279 (1971).
A. Aslanyan and E. B. Davies, “Spectral instability for some Schrödinger operators,” Numer. Math., 85, 525–552 (2000).
E. Balslev and J. M. Combes, “Spectral properties of many body Schrödinger operators with dilation: analytic interactions,” Commun. Math. Phys., 22, 280–294 (1971).
K. Kh. Boimatov, “Sturm–Liouville operators with matrix potentials,” Mat. Zametki, 16, No. 6, 921–932 (1974).
K. Kh. Boimatov, “Some spectral properties of matrix differential operators far from being selfadjoint,” Funkts. Anal. Prilozh., 29, No. 3, 55–58 (1995).
B. M. Brown and M. S. P. Eastham, “Spectral instability for some Schrödinger operators,” J. Comput. Appl. Math., 148, 49–63 (2002).
E. B. Davies, “Non-self-adjoint differential operators,” Bull. London Math. Soc., 34, No. 5, 513–532 (2002).
E. B. Davies, “Eigenvalues of an elliptic system,” Math. Z., 243, 719–743 (2003).
M. Giertz, “On the solution in L 2 of y″ + (λ − q(x))y = 0 when q is rapidly increasing,” Proc. London Math. Soc., 14, 53–73 (1964).
I. Ts. Gokhberg and M. G. Krein, Introduction to the Theory of Linear Non-Self-Adjoint Operators [in Russian], Nauka, Moscow (1965).
M. Hager, “Instabilité spectrale semiclassique d’operateurs,” Ann. H. Poincaré, 7, No. 6, 1035–1064 (2002).
Kh. K. Ishkin, “Asymptotics of the spectrum and the regularized trace of higher-order singular differential operators,” Differ. Uravn., 31, No. 10, 480–490 (1995).
Kh. K. Ishkin, “Necessary conditions for the localization of the spectrum of the Sturm–Liouville problem on a curve,” Mat. Zametki, 78, No. 1, 72–84 (2005).
Kh. K. Ishkin, “On localization of the spectrum of the problem with complex weight,” Fundam. Prikl. Mat., 12, No. 5, 49–64 (2006).
Kh. K. Ishkin, “On the spectral instability of the Sturm–Liouville operator with a complex potential,” Differ. Uravn., 45, No. 4, 480–495 (2009).
Kh. K. Ishkin, “Localization criterion for the spectrum of the Sturm–Liouville operator on a curve,” Algebra Anal., 28, No. 1, 52–88 (2016).
T. Kato, “Fractional powers of dissipative operators,” J. Math. Soc. Jpn., 13, 246–273 (1961).
T. Kato, “Fractional powers of dissipative operators, II,” J. Math. Soc. Jpn., 14, 242–248 (1962).
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin (1966).
M. V. Keldysh, “On eigenvalues and eigenfunctions of some classes of non-self-adjoint equations,” Dokl. Akad. Nauk SSSR, 77, No. 1, 11–14 (1951).
V. B. Lidskii, “Conditions of completeness of the system of root subspaces of non-self-adjoint operators with discrete spectra,” Tr. Mosk. Mat. Obshch., 8, 83–120 (1959).
V. B. Lidskii, “A non-self-adjoint operator of the Sturm–Liouville type with discrete spectrum,” Tr. Mosk. Mat. Obshch., 9, 45–79 (1960).
M. A. Naimark, Linear Differential Operators [in Russian], Nauka, Moscow (1969).
D. Ray, “On spectra of second-order differential operator,” Trans. Am. Math. Soc., 77, 299–321 (1954).
M. Reed and B. Simon, Methods of Modern mathematical Physics, Vol. 1, Functional Analysis, Academic Press, New York–London (1972).
J. Sjöstrand, Spectral instability for non-selfadjoint operators, preprint, Ecole Polytechnique, Palaiseau Cedex, France (2002).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 141, Differential Equations. Spectral Theory, 2017.
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Valiullina, L.G., Ishkin, K.K. On the Localization Conditions for the Spectrum of a Non-Self-Adjoint Sturm–Liouville Operator with Slowly Growing Potential. J Math Sci 241, 556–569 (2019). https://doi.org/10.1007/s10958-019-04445-0
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DOI: https://doi.org/10.1007/s10958-019-04445-0