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Summary:
In this article, we consider the operator $L$ defined by the differential expression \[ \ell (y)=-y^{\prime \prime }+q(x) y ,\quad - \infty < x < \infty \] in $L_2(-\infty ,\infty )$, where $q$ is a complex valued function. Discussing the spectrum, we prove that $L$ has a finite number of eigenvalues and spectral singularities, if the condition \[ \sup _{-\infty < x < \infty } \Big \lbrace \exp \bigl (\epsilon \sqrt{|x|}\bigr ) |q(x)|\Big \rbrace < \infty , \quad \epsilon > 0 \] holds. Later we investigate the properties of the principal functions corresponding to the eigenvalues and the spectral singularities.
References:
[1] Ö. Akin, E. Bairamov: On the structure of discrete spectrum of the non-selfadjoint system of differential equations in the first order. J. Korean Math. Soc. 32 (1995), 401–413. MR 1355662
[2] Yu. M. Berezanski: Expansion in Eigenfunctions of Selfadjoint Operators. Amer. Math. Soc., Providence R. I., 1968.
[3] B. B. Blashak: On the second-order differential operator on the whole axis with spectral singularities. Dokl. Akad. Nauk Ukr. SSR 1 (1966), 38–41. (Russian)
[4] M. G. Gasymov, F. G. Maksudov: The principal part of the resolvent of non-selfadjoint operators in neighbourhood of spectral singularities. Functional Anal. Appl. 6 (1972), 185–192. MR 0312307
[5] S. V. Hruscev: Spectral singularities of dissipative Schrödinger operators with rapidly decreasing potential. Indiana Univ. Math. Jour. 33 (1984), 613–638. DOI 10.1512/iumj.1984.33.33033 | MR 0749318 | Zbl 0548.34022
[6] M. V. Keldysh: On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators. Russian Math. Survey 26 (1971), 15–44. DOI 10.1070/RM1971v026n04ABEH003985 | Zbl 0225.47008
[7] V. E. Lyance: A differential operator with spectral singularities I, II. AMS Translations 60 (2) (1967), 185–225, 227–283.
[8] F. G. Maksudov, B. P. Allakhverdiev: Spectral analysis of a new class of non-selfadjoint operators with continuous and point spectrum. Soviet Math. Dokl. 30 (1984), 566–569.
[9] V. A. Marchenko: Sturm-Liouville Operators and Their Applications. Birkhauser, Basel, 1986. MR 0897106
[10] M. A. Naimark: Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint operator of the second order on a semi-axis. AMS Translations 16 (2) (1960), 103–193. MR 0117382
[11] M. A. Naimark: Linear Differential Operators II. Ungar, New York, 1968. MR 0353061
[12] B. S. Pavlov: The non-selfadjoint Schrödringer operator. Topics in Mathematical Physics 1 Consultants Bureau, New York (1967), 87–114.
[13] B. S. Pavlov: On separation conditions for the spectral components of a dissipative operator Math. USSR Izvestiya 9 (1975), 113–135. DOI 10.1070/IM1975v009n01ABEH001468
[14] I. I. Privalov: The Boundary Properties of Analytic Functions. Hochschulbücher für Math. Bb. 25, VEB Deutscher Verlag, Berlin, 1956.
[15] J. T. Schwartz: Some non-selfadjoint operators. Comm. Pure and Appl. Math. 13 (1960), 609–639. DOI 10.1002/cpa.3160130405 | MR 0165372 | Zbl 0096.08901
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