Inverse spectral problems for non-selfadjoint second-order differential operators with Dirichlet boundary conditions
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RESEARCH
Open Access
Inverse spectral problems for non-selfadjoint second-order differential operators with Dirichlet boundary conditions Sergey A Buterin1 , Chung-Tsun Shieh2* and Vjacheslav A Yurko1 *
Correspondence: [email protected] 2 Department of Mathematics, Tamkang University, Tamsui, New Taipei, 25137, Taiwan Full list of author information is available at the end of the article
Abstract We study the inverse problem for non-selfadjoint Sturm-Liouville operators on a finite interval with possibly multiple spectra. We prove the uniqueness theorem and obtain constructive procedures for solving the inverse problem along with the necessary and sufficient conditions of its solvability and also prove the stability of the solution. MSC: 34A55; 34B24; 47E05 Keywords: non-selfadjoint Sturm-Liouville operators; inverse spectral problems; method of spectral mappings; generalized spectral data; generalized weight numbers
1 Introduction Inverse spectral problems consist of recovering operators from given spectral characteristics. Such problems play an important role in mathematics and have many applications in natural sciences and engineering (see, for example, monographs [–] and the references therein). We study the inverse problem for the Sturm-Liouville operator corresponding to the boundary value problem L = L(q(x), T) of the form y := –y + q(x)y = λy, y() = y(T) = ,
< x < T < ∞,
() ()
where q(x) ∈ L (, T) is a complex-valued function. The results for the non-selfadjoint operator (), () that we obtain in this paper are crucial in studying inverse problems for Sturm-Liouville operators on graphs with cycles. Here also lies the main reason of considering the case of Dirichlet boundary conditions () and arbitrary length T of the interval. For the selfadjoint case, i.e., when q(x) is a real-valued function, the inverse problem of recovering L from its spectral characteristics was investigated fairly completely. As the most fundamental works in this direction we mention [, ], which gave rise to the so-called transformation operator method having become an important tool for studying inverse problems for selfadjoint Sturm-Liouville operators. The inverse problems for non-selfadjoint operators are more difficult for investigation. Some aspects of the inverse problem theory for non-selfadjoint Sturm-Liouville operators were studied in [–] and other papers. © 2013 Buterin et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Buterin et al. Boundary Value Problems 2013, 2013:180 http://www.boundaryvalueproblems.com/content/2013/1/180
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In the present paper, we use the method of spectral mappings [], which is effective for a wide class of differential and difference operators including non-selfadjoint ones. The method of spectral mappings is
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