Behaviors of Eigenvalues and Eigenfunctions of The Singular Shrödinger Operator

Yıl 2020, Cilt: 12 Sayı: 2, 151 - 156, 31.12.2020

Rauf Amirov , Sevim Durak

https://doi.org/10.47000/tjmcs.793631

Cited By: 1

Öz

Let us show the boundary value problem $L\left( q\right) $ with the $-y^{^{\prime\prime}}+q(x)y=\lambda y$ differential equation in the $\left[0,1\right] $ interval, and the $y(0)=0,y(1)=0$ boundary conditions in $\sigma\left( x\right) \equiv\int\limits_{0}^{x}q(t)dt.$ It is important to examine this operator as the solution to many problems of quantum physics is closely linked to the learning of the spectral properties of the operator $L\left( q\right) $. Singular Shr\"{o}dinger operators are characterized by the assumption that, in classical theory, the function $q(x)$ is not summable in the interval $\left[ a,b\right] $ for example it has singularity that cannot be integrated in at least one of the end points of the interval or at one of its internal points, or that the interval $\left( a,b\right) $ is infinite interval. 

In the present study, firstly, the operator of $L\left( q\right) $ will be proved to be well-defined in the class of distribution functions with first-order singularity, which is the larger class of functions. In the following step, the concepts of eigenvalue and eigenfunctions are defined for the well-defined $L\left( q\right) $ operator and the representations for their behaviour are obtained.

Anahtar Kelimeler

Singular Shrödinger Operator, eigenvalue, eigenfunction

Kaynakça

• Amirov, R.Kh., Guseinov, I.M., \textit{Boundary Value Problems for a class of Sturm-Liouville operators with Nonintegrable Potential}, Diff. Equations, \textbf{38}(8)(2002), 1195-1197.
• Amirov, R.Kh., \c{C}akmak, Y., G\"{u}lyaz, S., \textit{Boundary value problem for second-order differential equations with coulomb singularity on a finite interval}, Indian J. Pure Appl. Math., \textbf{37}(3)(2006), 125-140.
• Amirov, R., Ergun, A., Durak, S., \textit{Half inverse problems for the quadratic pencil of the Sturm-Liouville equations with impulse}, Numerical Methods for Partial Differential Equations, DOI: 10.1002/num.22559, (2020).
• Amirov, R.Kh., Ergun, A., \textit{Direct and inverse problems for diffusion operat\"{o}r with discontinuity points}, TWMS J. App. Eng. Math. \textbf{9}(1)(2019), 9-21.
• Ergun, A., \textit{Integral representation for solution of discontinuous diffusion operator with jump conditions}, Cumhuriyet Science Journal, \textbf{39}(4)(2018), 842-863.
• Naimark, M.A., Lineer differential operators, Moscow, Nauka, 1969.
• Savchuk, A.M., Shkalikov, A.A., \textit{Sturm-Liouville operators with singular potentials}, Math. Zametki, \textbf{66}(1999), 897-912.
• Shkalikov, A.A., \textit{Boundary value problems for the ordinary differential equations with the parameter in the boundary conditions}, Trudy Sem. I. G. Petrovskogo, \textbf{9}(1983), 190-229.