arXiv subjects
Publications and source records attributed to S. A. Avdonin.
We consider applications of the Boundary Control (BC) method to generalized spectral estimation problems and to inverse source problems. We derive the equations of the BC method for this problems and show that solvability of this equations crucially depends on the controllability properties of the corresponding dynamical system and properties of corresponding families of exponentials.
We link the Boundary Control Theory and the Titchmarsh-Weyl Theory. This provides a natural interpretation of the $A-$amplitude due to Simon and yields a new efficient method to evaluate the Titchmarsh-Weyl $m-$function associated with the Schrödinger operator $H=-\partial _{x}^{2}+q\left( x\right) $ on $L_{2}\left( 0,\infty \right) $ with Dirichlet boundary condition at $x=0.$
We establish connections between different approaches to inverse spectral problems: the classical Gelfand--Levitan theory, the Krein method, the Simon theory, the approach proposed by Remling and the Boundary Control method. We show that the Boundary Control approach provides simple and physically motivated proofs of the central results of other theories. We demonstrate also the connections between the dynamical and spectral data and derive the local version of the classical Gelfand--Levitan equations.
We consider the in-plane motion of elastic strings on tree-like network, observed from the 'leaves'. We investigate the inverse problem of recovering not only the physical properties i.e. the 'optical lengths' of each string, but also the topology of the tree which is represented by the edge degrees and the angles between branching edges. To this end use the boundary control method for wave equations established in~\cite{AK,B}. It is shown that under generic assumptions the inverse problem can be solved by applying measurements at all leaves, the root of the tree being fixed.
In this article the authors continue the discussion in \cite{ALM} about inverse problems for second order elliptic and hyperbolic equations on metric trees from boundary measurements. In the present paper we prove the identifiability of varying densities of a planar tree-like network of strings along with the complete information on the graph, i.e. the lengths of the edges, the edge degrees and the angles between neighbouring edges. The results are achieved using the Titchmarch-Weyl function for the spectral problem and the Steklov-Poincar{é} operator for the dynamic wave equation on the tree. The general result is obtained by a peeling argument which reduces the inverse problem layer-by-layer from the leaves to the clamped root of the tree.
We study the boundary control problems for the wave, heat, and Schrödinger equations on a finite graph. We suppose that the graph is a tree (i.e., it does not contain cycles), and on each edge an equation is defined. The control is acting through the Dirichlet condition applied to all or all but one boundary vertices. The exact controllability in $L_2$-classes of controls is proved and sharp estimates of the time of controllability are obtained for the wave equation. The null controllability for the heat equation and exact controllability for the Schrödinger equation in arbitrary time interval are obtained.
We consider the generalized spectral estimation problem in infinite dimensional spaces. We solve this problem using the boundary control approach to inverse theory and provide an application to the initial boundary value problem for a hyperbolic system.
Interest in inverse dynamical, spectral and scattering problems for differential equations on graphs is motivated by possible applications to nano-electronics and quantum waveguides and by a variety of other classical and quantum applications. Recently a new effective leaf peeling method has been proposed by S. Avdonin and P. Kurasov \cite{AK} for solving inverse problems on trees (graphs without cycles). It allows recalculating efficiently the inverse data from the original tree to the smaller trees, `removing' leaves step by step up to the rooted edge. In this paper we describe the main step of the spectral and dynamical versions of the peeling algorithm -- recalculating the inverse data for the `peeled tree'.
We study inverse boundary problems for a one dimensional linear integro-differential equation of the Gurtin--Pipkin type with the Dirichlet-to-Neumann map as the inverse data. Under natural conditions on the kernel of the integral operator, we give the explicit formula for the solution of the problem with the observation on the semiaxis $t>0.$ For the observation on finite time interval, we prove the uniqueness result, which is similar to the local Borg--Marchenko theorem for the Schrödinger equation.
We consider the inverse problem of the determining the potential in the dynamical Schrödinger equation on the interval by the measurement on the whole boundary. Provided that source is \emph{generic} using the Boundary Control method we recover the spectrum of the problem from the observation at either left or right end points. Using the specificity of the one-dimensional situation we recover the spectral function, reducing the problem to the classical one which could be treated by known methods. We adapt the algorithm to the situation when only the finite number of eigenvalues are known and provide the result on the convergence of the method.
We study the tunnelling trough a potential barrier of the system of two quantum correlated particles. The system is considered in one dimension. The interaction with the barrier and between particles is approximated by $δ$-potentials. Assuming that the particles have the same masses, we reduce the problem to the set of independent planar scattering problems corresponding to two-body flux symmetry with respect to the barrier. In order to solve the problems we apply the method of Sommerfeld-Maluzhinets integral transformation, which requires the solution of a set of functional equations. We show that the sub-problem which is antisymmetric with respect to the center of mass allows the solution in frames of the Bethe ansatz. So we give the exact solution of this problem. For the subproblem which is symmetric we describe the class of solutions for the functional equations and give an approach for obtaining the solution.
The problem of spin-dependent transport of electrons through a finite array of quantum dots attached to 1D quantum wire (spin gun) for various semiconductor materials is studied. The Breit-Fermi term for spin-spin interaction in the effective Hamiltonian of the device is shown to result in a dependence of transmission coefficient on the spin orientation. The difference of transmission probabilities for singlet and triplet channels can reach few percent for a single quantum dot. For several quantum dots in the array due to interference effects it can reach approximately 100% for some energy intervals. For the same energy intervals the conductance of the device reaches the value $\approx 1$ in $[e^{2}/π\hbar]$ units. As a result a model of the spin-gun which transforms the spin-unpolarized electron beam into completely polarized one is suggested.
Linear combinations of exponentials $e^{iλ_kt}$ in the case where the distance between some points $λ_k$ tends to zero are studied. D. Ullrich has proved the basis property of the divided differences of exponentials in the case when the groups consist of equal number of points all of them are close enough to integers. We have generalized this result for groups with arbitrary number of close points and obtained a full description of Riesz bases of exponential divided differences. The application to a control problem is presented.