On the spectral investigation of the scattering problem for some version of one-dimensional Schrödinger equation with turning point

Yüzüncü Yıl Üniversitesi Fen Bilimleri Enstitüsü Dergisi

2023

This study focuses on the spectral features of the non-hermitian singular operator with an out of the ordinary type weight function. Take into consideration the one-dimensional time dependent Schrödinger type differential equation -y^( ^'' )+q(x)y=μ^2 ρ(x)y,x∈[0,∞), holding the initial condition y(0)=0, and the density function defined with completely negative value as ρ(x)=-1. There are enormous number of the papers considering the positive values of ρ(x) for both continuous and discontinuous cases. The structure of the density function affects the analytical properties and representations of the solutions of the equation. Differently from the classical literature, we used the hyperbolic type representations of the fundamendal solutions of the equation to obtain the spectrum of the operator. Additionally, the requirements for finiteness of eigenvalues and spectral singularities were addressed. Hence, Naimark’s and Pavlov’s conditions were adopted to the negative density function case.

Section: Solutions To the Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Yüzüncü Yıl Üniversitesi Fen Bilimleri Enstitüsü Dergisi

2023

“…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…”

Section: Construction Of Scattering Solutions and Scattering Functionmentioning

confidence: 99%

“…$$ Thus,

S\left(x,{\mu}&amp;#x0005E;2\right)

and

C\left(x,{\mu}&amp;#x0005E;2\right)

can be represented by the hyperbolic type representations

\begin{array}{cc}\hfill S\left(x,{\mu}&amp;#x0005E;2\right)&amp;amp; &amp;#x0003D;\frac{\sinh \mu x}{\mu },\hfill \\ {}\hfill C\left(x,{\mu}&amp;#x0005E;2\right)&amp;amp; &amp;#x0003D;\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}&amp;#x0005E;2\right)

has the Volterra type integral representation as

S\left(x,{\mu}&amp;#x0005E;2\right)&amp;#x0003D;\frac{\sinh \mu x}{\mu }&amp;#x0002B;\int_0&amp;#x0005E;xB\left(x,t\right)\frac{\sinh \mu t}{\mu } dt,

and

C\left(x,{\mu}&amp;#x0005E;2\right)

has the Volterra type integral representation as

C ((), x, μ)

…”

Section: Construction Of Scattering Solutions and Scattering Functionmentioning

confidence: 99%

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

Math Methods in App Sciences

2023

This paper aims to investigate the scattering function and discrete spectrum of the impulsive Sturm‐Liouville type differential operator with a turning point and nonlinear eigenparameter‐dependent boundary condition. Using hyperbolic type representations of the fundamental solutions, the operator's discrete spectrum was constructed. We presented asymptotic equations for the fundamental solutions and the Jost function. Finally, we stated an example to demonstrate the paper's major points.

“…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].…”

Section: Introductionmentioning

confidence: 99%

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

Universal Journal of Mathematics and Applications

Görgülü

2023

In this paper, we shall study the spectral properties of the non-selfadjoint operator in the space $L_{\varrho }^{2}\left(\mathbb{R}_{+}\right) $ generated by the Sturm-Liouville differential equation \begin{equation*} -y^{^{\prime \prime }}+q\left( x\right) y=\omega ^{2}\varrho \left( x\right) y, \quad x \in \mathbb{R}_{+} \end{equation*} with the integral type boundary condition \begin{equation*} \int \limits_{0}^{\infty }G\left( x \right) y\left( x\right) dx+ \gamma y^{\prime }\left( 0\right) -\theta y\left( 0\right) =0 \end{equation*} and the non-standard weight function \begin{equation*} \varrho \left( x\right) =-1 \end{equation*} where $\left \vert \gamma \right \vert +\left \vert \theta \right \vert \neq 0$. There are an enormous number of papers considering the positive values of $ \varrho \left( x\right) $ for both continuous and discontinuous cases. The structure of the weight function affects the analytical properties and representations of the solutions of the equation. Differently from the classical literature, we used the hyperbolic type representations of the fundamental solutions of the equation to obtain the spectrum of the operator. Moreover, the conditions for the finiteness of the eigenvalues and spectral singularities were presented. Hence, besides generalizing the recent results, Naimark's and Pavlov's conditions were adopted for the negative weight function case.