The inverse problem for Schrödinger operators on metric graphs is investigated in the presence of magnetic field. Graphs without loops and with Euler characteristic zero are considered. It is shown that the knowledge of the Titchmarsh-Weyl matrix function (Dirichlet-to-Neumann map) for just two values of the magnetic field allows one to reconstruct the graph and potential on it provided a certa...

We examine two kinds of spectral theoretic situations: First, we recall the case of self-adjoint half-line Schrödinger operators on [a,∞), a ∈ R, with a regular finite end point a and the case of Schrödinger operators on the real line with locally integrable potentials, which naturally lead to Herglotz functions and 2 × 2 matrix-valued Herglotz functions representing the associated Weyl–Titchma...

We examine two kinds of spectral theoretic situations: First, we recall the case of self-adjoint half-line Schrödinger operators on [a,∞), a ∈ R, with a regular finite end point a and the case of Schrödinger operators on the real line with locally integrable potentials, which naturally lead to Herglotz functions and 2× 2 matrix-valued Herglotz functions representing the associated Weyl–Titchmar...

In the paper we propose a direct method for recovering Sturm–Liouville potential from Weyl–Titchmarsh m-function given on countable set of points. We show that using Fourier–Legendre series expansion transmutation operator integral kernel problem reduces to an infinite linear system equations, which is uniquely solvable if so original problem. The solution this allows one reconstruct characteri...

Relations between halfand full-lattice CMV operators with scalarand matrix-valued Verblunsky coefficients are investigated. In particular, the decoupling of full-lattice CMV operators into a direct sum of two half-lattice CMV operators by a perturbation of minimal rank is studied. Contrary to the Jacobi case, decoupling a full-lattice CMV matrix by changing one of the Verblunsky coefficients re...

We prove local and global versions of Borg–Marchenko-type uniqueness theorems for half-lattice and full-lattice CMV operators (CMV for Cantero, Moral, and Velázquez [15]). While our half-lattice results are formulated in terms of Weyl–Titchmarsh functions, our full-lattice results involve the diagonal and main off-diagonal Green’s functions.

We consider the dissipative singular q-Sturm-Liouville operators acting in Hilbert space L2 w,q(R+), that extensions of a minimal symmetric operator with deficiency indices (2, 2) (in limit-circle case).We construct self-adjoint dilation and its incoming outgoing spectral representations, which make it possible to determine scattering matrix terms Weyl-Titchmarsh function operator. also functio...

Let us write λ = z, where 0 ≤ arg z < π when 0 ≤ arg λ < 2π. Then (1. 3) implies that there is a solution ψ(x, z) of (1. 1) such that ψ(x, z) ∼ exp(izx), ψ′(x, z) ∼ iz exp(izx) (1. 4) as x → ∞, and ψ(x, z) is analytic in z for im z > 0 [7, Theorem 1.9.1]. Then ψ(x, z) is the Weyl L2(0,∞) solution of (1. 1) when λ is non-real and it forms the basis of the Weyl-Titchmarsh spectral theory of (1. 1...