Behaviors of Eigenvalues and Eigenfunctions of The Singular Shrödinger Operator

Abstract

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.