On the Inverse Problem of Scattering Theory for a Differential Operator of the Second Order

Mamedov, Khanlar R.; Menken, Hamza

doi:10.1016/s0304-0208(04)80167-1

Cited by 5 publications

(4 citation statements)

References 8 publications

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“…The inverse problem with a spectral parameter in the boundary condition was examined for different characteristics: from spectral function in to Weyl function in . In special cases, the inverse problem to the scattering data was studied in for the boundary condition . Spectral analysis for the boundary value problem with the spectral parameter appearing linearly in the boundary condition was dealt in .…”

Section: Introductionmentioning

confidence: 99%

“…We must give the following lemma, which could be proved like in . Lemma The identity

\begin{matrix} aligned \\ right \frac{2 iλφ MathClass-open(x, λ MathClass-close)}{(β_{0} + i β_{1} λ + β_{2} λ^{2}) e^{'} MathClass-open(0, λ MathClass-close) + (α_{0} + i α_{1} λ + α_{2} λ^{2}) e MathClass-open(0, λ MathClass-close)} \\ right = \overset{MathClass-op̃}{e} MathClass-open(x, λ MathClass-close) - S MathClass-open(λ MathClass-close) f MathClass-open(x, λ MathClass-close) \end{matrix}

holds for all real λ ≠ 0, where

S (λ) MathClass-rel= ((), β_{0} MathClass-bin+ i β_{1} λ MathClass-bin+ β_{2} λ^{2}) {\overset{e}{false}}^{MathClass-rel'} (0 MathClass-punc, λ) MathClass-bin+ ((), α_{0} MathClass-bin+ i α_{1} λ MathClass-bin+ α_{2} λ^{2}) \overset{e}{false} (0 MathClass-punc, <...>

…”

Section: Introductionmentioning

confidence: 99%

“…The function S ( λ ) is a meromorphic function on the upper half‐plane Im λ > 0, and the poles coincide with the zeros of E ( λ ). After examining the properties of the zeros of E ( λ ), we can say that the zeros of E ( λ ) may have only a finite number of zeros on the half‐plane Im λ > 0; they are all simple and lie on the imaginary axis (see ), where

E (λ) MathClass-rel= ((), β_{0} MathClass-bin+ i β_{1} λ MathClass-bin+ β_{2} λ^{2}) e^{MathClass-rel'} (0 MathClass-punc, λ) MathClass-bin+ ((), α_{0} MathClass-bin+ i α_{1} λ MathClass-bin+ α_{2} λ^{2}) e ((), 0 MathClass-punc, λ) MathClass-punc.

…”

Section: Introductionmentioning

confidence: 99%

“…This integral equation admits a unique solution K ( x , t ) for each fixed x ≥ 0 in the space L 1 ( x , ∞ ). Note that in special cases, the aforementioned relations were obtained in .…”

Section: Introductionmentioning

confidence: 99%

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Continuity of the scattering function and Levinson‐type formula of Klein–Gordon equation

Karaman

Koşar

2012

Math Methods in App Sciences

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Communicated by A. KirschWe considered the inverse problem of scattering theory for a boundary value problem on the half line generated by Klein-Gordon differential equation with a nonlinear spectral parameter-dependent boundary condition. We defined the scattering data, and we proved the continuity of the scattering function S. /; in a special case, the relation for the difference of the logarithm of the scattering function, which is called the Levinson-type formula, was obtained. Copyright

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Section: Introductionmentioning

confidence: 99%

“…We must give the following lemma, which could be proved like in . Lemma The identity

\begin{matrix} aligned \\ right \frac{2 iλφ MathClass-open(x, λ MathClass-close)}{(β_{0} + i β_{1} λ + β_{2} λ^{2}) e^{'} MathClass-open(0, λ MathClass-close) + (α_{0} + i α_{1} λ + α_{2} λ^{2}) e MathClass-open(0, λ MathClass-close)} \\ right = \overset{MathClass-op̃}{e} MathClass-open(x, λ MathClass-close) - S MathClass-open(λ MathClass-close) f MathClass-open(x, λ MathClass-close) \end{matrix}

holds for all real λ ≠ 0, where

S (λ) MathClass-rel= ((), β_{0} MathClass-bin+ i β_{1} λ MathClass-bin+ β_{2} λ^{2}) {\overset{e}{false}}^{MathClass-rel'} (0 MathClass-punc, λ) MathClass-bin+ ((), α_{0} MathClass-bin+ i α_{1} λ MathClass-bin+ α_{2} λ^{2}) \overset{e}{false} (0 MathClass-punc, <...>

…”

Section: Introductionmentioning

confidence: 99%

E (λ) MathClass-rel= ((), β_{0} MathClass-bin+ i β_{1} λ MathClass-bin+ β_{2} λ^{2}) e^{MathClass-rel'} (0 MathClass-punc, λ) MathClass-bin+ ((), α_{0} MathClass-bin+ i α_{1} λ MathClass-bin+ α_{2} λ^{2}) e ((), 0 MathClass-punc, λ) MathClass-punc.

…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Continuity of the scattering function and Levinson‐type formula of Klein–Gordon equation

Karaman

Koşar

2012

Math Methods in App Sciences

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Inverse scattering problem for Sturm-Liouville operator with nonlinear dependence on the spectral parameter in the boundary condition

Mamedov

Koşar

2010

Math. Meth. Appl. Sci.

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The inverse problem of the scattering theory for Sturm-Liouville operator on the half line with boundary condition depending quadratic on the spectral parameter is considered. Scattering data are defined, some properties of the scattering data are examined, the main equation is obtained, solvability of the integral equation is proved and uniqueness of algorithm to the potential with given scattering data is studied.

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Uniqueness of the solution to the inverse problem of scattering theory for spectral parameter polynomially dependent Kleın-Gordon s-wave equation

Arpat¹,

Yokus²

2023

Adv. Studies: Euro-Tbilisi Math. J.

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On the Inverse Problem of Scattering Theory for a Differential Operator of the Second Order

Cited by 5 publications

References 8 publications

Continuity of the scattering function and Levinson‐type formula of Klein–Gordon equation

Continuity of the scattering function and Levinson‐type formula of Klein–Gordon equation

Inverse scattering problem for Sturm-Liouville operator with nonlinear dependence on the spectral parameter in the boundary condition

Uniqueness of the solution to the inverse problem of scattering theory for spectral parameter polynomially dependent Kleın-Gordon s-wave equation

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