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Asymptotic analysis of non-self-adjoint Hill operators

100%
Veliev O.
Open Mathematics
|
2013
|
tom 11
|
nr 12
2234-2256
EN
We obtain uniform asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators L t (q) with a potential q ∈ L 1[0,1] and t-periodic boundary conditions, t ∈ (−π, π]. Using these formulas, we find sufficient conditions on the potential q such that the number of spectral singularities in the spectrum of the Hill operator L(q) in L 2(−∞,∞) is finite. Then we prove that the operator L(q) has no spectral singularities at infinity and it is an asymptotically spectral operator provided that the potential q satisfies sufficient conditions.
2
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Inverse problems on star-type graphs: differential operators of different orders on different edges

76%
Yurko V.
Open Mathematics
|
2014
|
tom 12
|
nr 3
483-499
EN
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.
3
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Ulam stability for a delay differential equation

76%
Otrocol D., Ilea V.
Open Mathematics
|
2013
|
tom 11
|
nr 7
1296-1303
EN
We study the Ulam-Hyers stability and generalized Ulam-Hyers-Rassias stability for a delay differential equation. Some examples are given.
4
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An inverse problem for Sturm-Liouville operators on the half-line having Bessel-type singularity in an interior point

64%
Fedoseev A.
Open Mathematics
|
2013
|
tom 11
|
nr 12
2203-2214
EN
We study the inverse problem of recovering Sturm-Liouville operators on the half-line with a Bessel-type singularity inside the interval from the given Weyl function. The corresponding uniqueness theorem is proved, a constructive procedure for the solution of the inverse problem is provided, also necessary and sufficient conditions for the solvability of the inverse problem are obtained.
5
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Numerical solution of inverse spectral problems for Sturm-Liouville operators with discontinuous potentials

64%
Efremova L., Freiling G.
Open Mathematics
|
2013
|
tom 11
|
nr 11
2044-2051
EN
We consider Sturm-Liouville differential operators on a finite interval with discontinuous potentials having one jump. As the main result we obtain a procedure of recovering the location of the discontinuity and the height of the jump. Using our result, we apply a generalized Rundell-Sacks algorithm of Rafler and Böckmann for a more effective reconstruction of the potential and present some numerical examples.
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