Linear Differential Operators Naimark.pdf
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The original form of symmetric boundary conditions for Lagrange symmetric (formally self-adjoint) ordinary linear differential expressions, was obtained by I.M. Glazman in his seminal paper of 1950. This result described all self-adjoint differential operators, in the underlying Hilbert function space, generated by real even-order differential expressions.
During his service in World War II Naimark wrote several papers on seismology, and helping to develop the Spectral theory of ordinary differential equations. He worked especially on second-order singular differential operators with a continuous spectrum, using eigenfunctions to describe their spectral decompositions, and studying the concept of a spectral singularity. His results are summarized in the monograph Linear Differential Operators, which was published in 1954.
In this paper we investigate certain aspects of the multiparameter spectral theory of systems of singular ordinary differential operators. Such systems arise in various contexts. For instance, separation of variables for a partial differential equation on an unbounded domain leads to a multiparameter system of ordinary differential equations, some of which are defined on unbounded intervals. The spectral theory of systems of regular differential operators has been studied in many recent papers, e.g. [1, 3, 6, 9, 19, 21], but the singular case has not received so much attention. Some references for the singular case are [7, 8, 10, 13, 14, 18, 20], in addition general multiparameter spectral theory for self adjoint operators is discussed in [3, 9, 19].
@article{VyacheslavYurko2014,abstract = {We study inverse spectral problems for ordinary differential equations on compact star-type graphs when differential equations have different orders on different edges. As the main spectral characteristics we introduce and study the so-called Weyl-type matrices which are generalizations of the Weyl function (m-function) for the classical Sturm-Liouville operator. We provide a procedure for constructing the solution of the inverse problem and prove its uniqueness.},author = {Vyacheslav Yurko},journal = {Open Mathematics},keywords = {Geometrical graphs; Differential operators; Inverse spectral problems; Weyl-type matrices; inverse spectral problems; differential operators on graphs; Weyl function},language = {eng},number = {3},pages = {483-499},title = {Inverse problems on star-type graphs: differential operators of different orders on different edges},url = { },volume = {12},year = {2014},}
TY - JOURAU - Vyacheslav YurkoTI - Inverse problems on star-type graphs: differential operators of different orders on different edgesJO - Open MathematicsPY - 2014VL - 12IS - 3SP - 483EP - 499AB - We study inverse spectral problems for ordinary differential equations on compact star-type graphs when differential equations have different orders on different edges. As the main spectral characteristics we introduce and study the so-called Weyl-type matrices which are generalizations of the Weyl function (m-function) for the classical Sturm-Liouville operator. We provide a procedure for constructing the solution of the inverse problem and prove its uniqueness.LA - engKW - Geometrical graphs; Differential operators; Inverse spectral problems; Weyl-type matrices; inverse spectral problems; differential operators on graphs; Weyl functionUR - ER -
In this paper we find a class of boundary conditions which determine dissipative differential operators of order three and prove that these operators have no real eigenvalues. The completeness of the system of eigenfunctions and associated functions is also established. 153554b96e
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