In this paper, we consider the inverse spectral problem for the impulsive Sturm-Liouville differential pencils on [0, π] with the Robin boundary conditions and the jump conditions at the point 2. We prove that two potentials functions on the whole interval and the parameters in the boundary and jump conditions can be determined from a set of eigenvalues for two cases: (i) the potentials given on where is the spectral parameter, p(x) ∈ W 1 2 [0, ], q(x) ∈ L 2 [0, ] are real-valued functions, is a real number, and > 0, ≠ 1. Here we denote by W m 2 [0, ] the space of functions f (x), x ∈ [0, ], such

Let us show the boundary value problem L (q) with the −y + q(x)y = λy differential equation in the [0, 1] interval, and the yIt 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 (q). Singular Shrödinger operators are characterized by the assumption that, in classical theory, the function q(x) is not summable in the interval [a, b] 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 (a, b) is infinite interval.In the present study, firstly, the operator of L (q) 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 (q) operator and the representations for their behaviour are obtained.

In this study, a diffusion operator is investigated on a star graph with nonhomogeneous edges. First, the behaviors of sufficiently large eigenvalues are learned, and then the solution of the inverse problem is given to determine the potential functions and parameters of the boundary condition on the star graph with the help of a dense set of nodal points and to obtain a constructive solution to the inverse problems of this class.

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In this paper, we study the inverse spectral problem for the quadratic differential pencils with discontinouty coefficient on [0, π] with seperably boundary conditions and the impulsive conditions at the point x = π 2 . We prove that two potentials functions on the interval [0, π] and the parameters in the boundary and impulsive conditions can be determined from a sequence of eigenvalues for two cases: (i) The potentials is given onThe potentials is given on ( π) , where 0 < α < 1 , respectively.

In this study, we consider the domains of the minimal and maximal operators generated of singular differential-expression-type Sturm-Liouville and obtain all self-adjoint extensions of the operator in terms of boundary conditions.