Behaviors of Eigenvalues and Eigenfunctions of The Singular Shrödinger Operator

Authors : Rauf AMİROV, Sevim DURAK

Pages : 151-156

Doi:10.47000/tjmcs.793631

View : 20 | Download : 8

Publication Date : 2020-12-31

Article Type : Research Paper

Abstract :Let us show the boundary value problem $L\leftinsert ignore into journalissuearticles values( q\right); $ with the $-y^{^{\prime\prime}}+qinsert ignore into journalissuearticles values(x);y=\lambda y$ differential equation in the $\left[0,1\right] $ interval, and the $yinsert ignore into journalissuearticles values(0);=0,yinsert ignore into journalissuearticles values(1);=0$ boundary conditions in $\sigma\leftinsert ignore into journalissuearticles values( x\right); \equiv\int\limits_{0}^{x}qinsert ignore into journalissuearticles values(t);dt.$ 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\leftinsert ignore into journalissuearticles values( q\right); $. Singular Shr\`{o}dinger operators are characterized by the assumption that, in classical theory, the function $qinsert ignore into journalissuearticles values(x);$ 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 $\leftinsert ignore into journalissuearticles values( a,b\right); $ is infinite interval.  In the present study, firstly, the operator of $L\leftinsert ignore into journalissuearticles values( 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\leftinsert ignore into journalissuearticles values( q\right); $ operator and the representations for their behaviour are obtained.
Keywords : Singular Shrödinger Operator, eigenvalue, eigenfunction