Inverse spectral and inverse nodal problems for Sturm-Liouville equations with point $\delta$ and $\delta'$-interactions

Abstract

Inverse spectral and inverse nodal problems are studied for Sturm-Liouville equations with point δ and δ-interactions. Uniqueness theorems are proved and a constructive procedure for the solutions is provided.

Spectral Properties and Inverse Nodal Problems for Singular Diffusion Equation

“…In addition, using the methods used in papers [5], [7], the following propositions are proved: Let ∆ 0 (λ) be the characteristic function of the problem corresponding to the case is q(x) ≡ 0 problem (1.1)- (1.3). In this case, it becomes ∆ 0 (λ) = ϕ 0 (π, λ) + Hϕ 0 (π, λ) , (2.3) where ϕ 0 (x, λ) is the solution of the equation (1.4), satisfying initial conditions (1.2) and discontinuity condition (1.5).…”

“…2 [0, 1] : y (x 0 + 0) − y (x 0 − 0) = ay (x 0 ) , x 0 ∈ (0, 1) ; y(0) = 0 = y (1) and is expressed by the differential operator given as L o = − d 2 dx 2 in Hilbert space L 2 [0, 1]. There is detailed information about the correct ( regular) definition of such operators and the examination of their spectral properties in [2,7,11] studies.…”

Inverse Nodal Problems for Pencils of Singular Sturm-Liouville Operators

“…2 [0, 1] : y ′ (x 0 +) − y ′ (x 0 −) = ay (x 0 ) , x 0 ∈ (0, 1) ; y(0) = 0 = y (1) and is expressed by the differential operator given as L o = − d 2 dx 2 in Hilbert space L 2 [0, 1]. There is detailed information about the correct (regular) definition of such operators and the examination of their spectral properties in [2], [14], [19], [20] studies.…”

Inverse nodal problems for singular diffusion equation

“…} and is expressed by the differential operator given as L 0 = − d 2 dx 2 in Hilbert space L 2 [0, 1] . There is detailed information about the correct ( regular) definition of such operators and the examination of their spectral properties [2][3][4].…”