Quantum graphs having one cycle are considered. It is shown that if the cycle contains at least three vertices, then the potential on the graph can be uniquely reconstructed from the corresponding Titchmarsh-Weyl function (Dirichlet-to-Neumann map) associated with graph’s boundary, provided certain non-resonant conditions are satisfied. PACS numbers: 03.65.Nk, 73.63.-b, 85.35.-p.

We systematically develop Weyl–Titchmarsh theory for singular differential operators on arbitrary intervals (a, b) ⊆ R associated with rather general differential expressions of the type τf = 1 r ( − ( p[f ′ + sf ] )′ + sp[f ′ + sf ] + qf ) , where the coefficients p, q, r, s are real-valued and Lebesgue measurable on (a, b), with p 6= 0, r > 0 a.e. on (a, b), and p−1, q, r, s ∈ Lloc((a, b); dx...

We systematically develop Weyl–Titchmarsh theory for singular differential operators on arbitrary intervals (a, b) ⊆ R associated with rather general differential expressions of the type

We show that the half-line m functions associated with vector-valued discrete Schrödinger operators are elements in Siegel upper half space. introduce a metric on space of to these operators. Then, we action transfer matrices is distance decreasing.

We develop singular Weyl–Titchmarsh–Kodaira theory for Jacobi operators. In particular, we establish existence of a spectral transformation as well as local Borg–Marchenko and Hochstadt–Liebermann type uniqueness results.

In the setting of adjoint pairs of operators we consider the question: to what extent does the Weyl M -function see the same singularities as the resolvent of a certain restriction AB of the maximal operator? We obtain results showing that it is possible to describe explicitly certain spaces S and S̃ such that the resolvent bordered by projections onto these subspaces is analytic everywhere that...

and Applied Analysis 3 and spectral theory for the system 1.2 . Shi studied Weyl-Titchmarsh theory and spectral theory for the system 1.2 in 33, 34 ; Clark and Gesztesy established the Weyl-Titchmarsh theory for a class of discrete Hamiltonian systems that include system 1.2 23 . Sun et al. established the GKN-theory for the system 1.2 35 . 1.3. Dynamic Equations A time scale T is an arbitrary ...