We consider the self-adjoint Smilansky Hamiltonian H_ξ in L²(R²) associated with the formal differential expression -∂²_x – ½(∂²_y +y²)- √2 ξyδ(x) in the sub-critical regime, ξ ϵ(0, 1). We demonstrate the existence of resonances for H_ξ on a countable subfamily of sheets of the underlying Riemann surface whose distance from the physical sheet is finite. On such sheets, we find resonance free regions and characterize resonances for small ξ>0. In addition, we refine the previously known results on the bound states of H_ξ in the weak coupling regime (ξ→ 0+). In the proofs we use Birman-Schwinger principle for H_ξ ,elements of spectral theory for Jacobi matrices, and the analytic implicit function theorem.

In this note, we continue our analysis (started in [1]) of the isotropic three-dimensional harmonic oscillator perturbed by a pair of identical attractive point interactions symmetrically situated with respect to the origin, that is to say, the mathematical model describing a symmetric quantum dot with a pair of point impurities. In particular, by making the coupling constant (to be renormalized) dependent also upon the separation distance between the two impurities, we prove that it is possible to rigorously define the unique self-adjoint Hamiltonian that, differently from the one introduced in [1], behaves smoothly as the separation distance between the impurities shrinks to zero. In fact, we rigorously prove that the Hamiltonian introduced in this note converges in the norm-resolvent sense to that of the isotropic three-dimensional harmonic oscillator perturbed by a single attractive point interaction situated at the origin having double strength, thus making this three dimensional model more similar to its one-dimensional analog (not requiring the renormalization procedure) as well as to the three-dimensional model involving impurities given by potentials whose range may even be physically very short but different from zero. Moreover, we show the manifestation of the Zeldovich effect, known also as level rearrangement, in the model investigated herewith. More precisely, we take advantage of our renormalization procedure to demonstrate the possibility of using the concept of ‘Zeldovich spiral’, introduced in the case of perturbations given by rapidly decaying potentials, also in the case of point perturbations.

Carbon nano-cluster cathodes exhibit a low threshold electron emission, which is 2–3 orders lower than on metals and semiconductors. We confirm the effect by direct experiments with graphene structures. We are suggesting a model based on the interference electrons wave function in 3D-space charge region of carbon structure interface with vacuum. The low-threshold emission is explained, in frames of the model, by the resonance properties of the barrier formed on the interface. Also in the following topics: interpretation of recent experimental findings for saturation of the field emission; local spectral analysis of multidimensional periodic lattices: dispersion via DN-map; examples of iso-energetic surfaces associated with solvable models of periodic lattice; Lagrangian version of the operator extension algorithm; solvable models of selected one-body spectral problems; quantum dot attached to the node of a quantum graph; a solvable model of a discrete lattice and spectral structure of a 1D superlattice via analytic perturbation procedure.

We consider the control and inverse problems for serially connected and tree-like networks of strings with point masses loaded at the internal vertices. We prove boundary controllability of the systems and the identifiability of varying coefficients of the string equations along with the complete information on the graph, i.e. the loaded masses, the lengths of the edges and the topology (connectivity) of the graph. The results are achieved using the Titchmarch-Weyl function for the spectral problem and the Steklov-Poincare operator for the dynamic wave equation on the tree. The general result is obtained by the leaf peeling method which reduces the inverse problem layer-by-layer from the leaves to the fixed root of the tree.

We consider the inverse problem for the dynamical system with discrete Schrödinger operator and discrete time. As inverse data, we take a response operator, the natural analog of the dynamical Dirichlet-to-Neumann map. We derive two types of equations of inverse problem and answer a question on the characterization of the inverse data, i.e. we describe the set of operators, which are response operators of the dynamical system governed by the discrete Schrödinger operator.

We study the direct and inverse eigenvalue problems for a pair of harmonic functions with a spectral parameter in boundary and coupling conditions. The direct problem is relevant to sloshing frequencies of free oscillations of a two-layer fluid in a container. The upper fluid occupies a layer bounded above by a free surface and below by a layer of fluid of greater density. Both fluids are assumed to be inviscid, incompressible, and heavy, whereas the free surface and the interface between fluids are considered bounded.

An algorithm and code for spectrum calculation for periodic nanostructures in homogeneous magnetic field are developed. The approach is based on the zero-range potentials model. The mathematical background of the model is based on the theory of self-adjoint extensions of symmetric operators.

We describe the analytic solution of the Cauchy problem for some fourth-order linear hyperbolic equations with constant coefficients in a half-plane in the case of two independent variables, assuming certain conditions for the coefficients. Suitable conditions are assumed for the coefficients, and the equation operator is composed of first-order linear operators.

A system of two arbitrary quantum particles moving on d-dimensional lattice interacting via some attractive potential is considered. The number of eigenvalues of the family h(k) is studied depending on the interaction energy of particles and the total quasi-momentum k ϵT^d (T^d - d-dimensional torus). Depending on the interaction energy, the conditions for h(0) that has simple or multifold virtual level at 0 are found.

 In this paper we give a characterization of normal subgroups for the group representation of the Cayley tree.

In this paper, we consider a model with logarithmical potential and with the set [0, 1] of spin values, on a Cayley tree Г^k of the order k. In the case k = 2, 3, we shall prove that, there is a unique translation-invariant splitting Gibbs measure for this model. For the case k = 4, we show that there are three translation-invariant Gibbs measures for this model.