The Nonself-Adjoint Schroedinger Operator. III
The present article contains a detailed derivation of some results previously summarized in a note by the author . It is also a continuation of the article published in the second volume of the present series . An example of a nonself-adjoint Schroedinger operator with a rapidly decreasing potential and an infinite number of eigenvalues was given in . Here, we will show that the spectrum of the Schroedinger operator can have a very complicated structure. Namely, there exist operators of this type whose eigenvalues possess a continuum of accumulation points. These results have been formulated as Theorems I, II, and III.
KeywordsDifferential Operator Unit Circle Real Axis Spectral Function Auxiliary Function
Unable to display preview. Download preview PDF.
- 1.B. S. Pavlov, “The spectral theory of nonself-adjoint operators,” Dokl, Akad. Nauk SSSR, Vol. 146, No. 6 (1962).Google Scholar
- 2.P. S. Pavlov, “The nonself-adjoint Schroedinger operator. II,” in: Topics in Mathematical Physics, Vol. 2, Consultants Bureau, New York (1968).Google Scholar
- 3.H. Weyl, “Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Functionen,” Math. Ann., Vol. 68 (1910).Google Scholar
- 4.E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Vol. I, Oxford University Press (1962).Google Scholar
- 5.V. A. Marchenko, “Expansions in eigenfunctions of nonself-adjoint second-order singular differential operators,” Matem. Sborn., 52 (94): 2 (1960).Google Scholar
- 6.B. S. Pavlov, “The nonself-adjoint Schroedinger operator,” in: Topics in Mathematical Physics, Vol. 1, Consultants Bureau, New York (1967).Google Scholar
- 7.I. I. Privalov, The Boundary Properties of Analytic Functions, GITTL (1950).Google Scholar
- 8.S. Warschawski, “On the differentiability at the boundary in conformal mapping,” Proc. Am. Math. Soc., Vol. 12, No. 4 (1961).Google Scholar
- 9.L. Carleson, “Sets of uniqueness for functions regular in the unit circle,” Acta Math., Vol. 87, No. 3–4 (1952).Google Scholar
- 10.I. M. Gel’fand and B. M. Levitan, “The construction of a differential equation from its spectral function,” Izv. Akad. Nauk SSSR, Ser. Math., Vol. 15 (1951).Google Scholar