The inverse scattering problem of some Schrödinger type equation with turning point

Abstract

In this paper the inverse scattering problem is considered for a version of the one-dimensional Schrödinger equation with turning point on the half-line (0, ∞). The scattering data of the problem is defined and the fundamental equation is derived. With the help of the derived fundamental equation, in terms of the scattering data, the potential is recovered uniquely.MSC: 58C40; 34L25; 34B05; 47A40

Scattering properties of the nonlinear eigenvalue‐dependent Sturm‐Liouville equations with sign‐alternating weight and jump condition

“…Also, both of the functions S and C are entire with regard to the variable 𝜇. Existence and uniqueness results of the solutions S ( x, 𝜇 2 ) and C ( x, 𝜇 2 ) can also be proven analogous to [2,[18][19][20]. Further, the Wronskian of the solutions S and C might be formulated as…”

“…$$ Thus, $$$ S\left(x,{\mu}^2\right) $$$ and $$$ C\left(x,{\mu}^2\right) $$$ can be represented by the hyperbolic type representations $$$$ {\displaystyle \begin{array}{cc}\hfill S\left(x,{\mu}^2\right)& =\frac{\sinh \mu x}{\mu },\hfill \\ {}\hfill C\left(x,{\mu}^2\right)& =\cosh \mu x.\hfill \end{array}} $$$$ Using the results of [2,18–20] and the constant coefficients method, one can easily verify that the fundamental solution $$$ S\left(x,{\mu}^2\right) $$$ has the Volterra type integral representation as $$$$ S\left(x,{\mu}^2\right)=\frac{\sinh \mu x}{\mu }+\int_0^xB\left(x,t\right)\frac{\sinh \mu t}{\mu } dt, $$$$ and $$$ C\left(x,{\mu}^2\right) $$$ has the Volterra type integral representation as …”

On the Spectrum of the Non-Selfadjoint Differential Operator with an Integral Boundary Condition and Negative Weight Function

“…The utility stemmed from the interconnection of studies on direct and inverse problems with the methods of solving many problems in mathematical analysis, keeps this research area vigorous [3][4][5][6][7]. This productive and efficient subject area, originated by the pioneer work of Naimark dealing with the singular non-self-adjoint problem for ρ(x) = 1, finds itself specialized sub-areas governing different but connected techniques, for example, cases considering positive weight [8][9][10][11][12][13], non-continuous weight [14][15][16][17], sign-changing weight [18][19][20] as well as discrete cases [21][22][23][24][25][26][27][28]. Especially, the spectral singularities of the non-selfadjoint problem under the integral boundary condition has been investigated in [9,10].…”

Negatif Yoğunluk Fonksiyonuna Sahip Kendine Eşlenik Olmayan Schrödinger Operatörü Üzerine Bir Çalışma

“…Unlike the known literature, inverse scattering and inverse spectral theory of the Sturm-Liouville type operators with sign-changing density function has been studied by Gasymov and El-Reheem (1993). The interested reader may also consult the papers (El-Raheem and Nasser, 2014;El-Raheem & Salama, 2015) and the references therein for the detailed information about the sign-valued density function case and its application in physics. The most crucial reason distinguishing this problem from the positive-valued weight function case is the new analytical difficulties that arising from the weight function's negative value.…”