B. S. Pavlov

Scattering problems on noncompact graphs

A Sturm-Liouville problem on compact graphs is formulated and analyzed. The scattering problem for the Schroedinger equation on noncompact graphs is also formulated and analyzed.

Two‐body scattering on a graph and application to simple nanoelectronic devices

The two‐body system on a graph with one junction is considered. The effective three‐body scattering problem turns out to be exactly solvable for pointwise interactions. Additional degrees of freedom corresponding to a dynamics of some structure (e.g., an atomic cluster) located in the junction (point of common contact) of three thin electrodes are considered. These degrees of freedom bring effective energy‐dependent interaction into the effective Schrodinger equation in the scattering channel. The wave function of the system is constructed in the explicit form using the extension theory methods. The obtained results are applied to the qualitative description of a simple three‐electrode nanoelectronic device. The perturbation theory approach based on the analysis of the Liouville equation is suggested for calculation of the conductivity for such a device in terms of the obtained wave function.

S-Matrix and Dirichlet-to-Neumann Operators

Publisher Summary This chapter provides an introduction to the Dirichlet-to-Neumann (DN) map. The DN map appears as an important detail in constructions of solutions of various inverse problems. The general features of the DN map for the Schrodinger operator L, which appears as a result of transformation of the conductance equation, are described. The original algorithm for the solution of the boundary measurements inverse problem is actually based on linearization of a map near the constant conductivity γ = 1 and is practically confined to the proof of uniqueness of the inverse map. In scattering problems, one needs to evaluate the DN map on the real axis of the spectral parameter γ. The resolvents of two operators A, B in Hilbert space are connected at the common regular point γ by the formula that is conveniently used to derive the Lippmann–Schwinger equation for Green functions and scattered waves of Schrodinger operators, since the perturbation of the potential is additive.

Extended Hilbert space approach to few‐body problems

A general formulation of the quantum scattering theory for a system of few particles, which have an internal structure, is given. Due to freezing out the internal degrees of freedom in the external channels, a certain class of energy‐dependent potentials is generated. By means of potential theory, a modified Faddeev equation is derived both in external and internal channels. The Fredholmity of these equations is proven and this is what provides a sound basis for solving the addressed scattering problem.

Scattering on a Compact Domain with Few Semi‐Infinite Wires Attached: Resonance Case

A Scattering problem is studied for Neumann Laplacian with a continuous potential on a domain with a smooth boundary and few semi-infinite wires attached to it at the points of contact on the boundary. In resonance case when the frequency of the incoming waves in the wires coincides with some resonance frequency of the domain the approximate formula for the transmission coefficient from one wire to another is derived: in the case of weak interaction between the domain and the wires the transmission coefficient is proportional to the product of values of the corresponding resonance eigenfunction of inner problem at the corresponding points of contact.

A star-graph model via operator extension

One usually assumes that the Schrodinger operator on a thin quantum network (“fattened graph”) can be simulated by the ordinary Schrodinger operator on the corresponding one-dimensional quantum graph. On the other hand, each quantum graph can be constructed of standard elements – star graphs. We prove that a thin star-shaped quantum network which consists of a compact domain and a few straight semi-infinite quantum wires, with a two-dimensional Schrodinger operator on it, can be simulated by the corresponding solvable model in form of a one-dimensional star graph: an outer space, with an ordinary Schrodinger operator on the leads, a resonance vertex supplied with an inner space and a finite matrix in it and an appropriate boundary condition connecting the inner and outer components of elements from the domain of the model. The model is (locally) quantitatively consistent: the scattering matrix of the model on a certain spectral interval serves an approximation of the scattering matrix of the network. The role of the constructed star-graph model as a “jump-start” in analytic perturbation procedure on continuous spectrum is discussed.

About Scattering on the Ring

The mathematical model of a simplest quasi-one-dimensional quantum network constructed of relatively narrow waveguides (the width of the waveguide is less than the de Broghlie wavelength of the electron in the material) is developed. This model allows to reduce the problem of calculating the current through the quantum network to the construction of scattered waves for some Schrodinger equation on the corresponding one-dimensional graph. We consider a graph consisting of a compact part and few semiinfinite rays attached to it via some boundary condi-tion depending on a parameter β (analog of the inverse exponential “mass” e -bH of a potential barrier H separating the rays from the compact part of the graph). This parameter regulates the connection between the rays and the compact part. Spectral properties of the Schrodinger operator on this graph are described with a special emphasis on the resonance case when the Fermi level in the rays coincides with one of eigenvalues of the nonperturbed Schrodinger operator on the ring. An explicit expression is obtained for the scattering matrix in the resonance case for weakening connection β → 0 between the rays and the compact part.

Low-threshold field emission from carbon nano-clusters

Detonation carbon materials (DCM) composed of non-equilibrium nano-structures show the low-threshold field emission (LTFE). These materials have forward-looking application especially due to high reproducibility of the LTFE-phenomenon on a surface of emitter, where the emitting centers are homogeneously distributed. In this paper we link the effect of LTFE to the nature of the corresponding wave functions based on the experiment results obtained for DCM by the field effect on electrolytes. As we had shown before DCM had been described by an ultra-relativistic dispersion function with extremely small effective mass of electrons and the size-quantization effect had been observed in DCM at even room temperature. Our results based on emission and electrolyte technics of the field-effect measurements in DCM along with modern observations of the field emission in strong electric fields allowed to propose a new resonance transmission model for LTFE-phenomenon, which is alternative to most widely discussed models based on the field-enhancing factors or barrier-lowering mechanisms.

Quantum scattering on a Cantor bar

A solvable model of the scattering of a one‐dimensional particle by the familiar ‘‘middle‐third’’ Cantor set is considered. It is shown that the presence of positive energy levels in such a model is typical. From the high‐energy behavior of the scattering, data computation of the Hausdorff dimension of the scatterer is suggested.

Spectral properties of a charged particle in antidot array: A limiting case of quantum billiard

A model of the periodic array of quantum antidots in the presence of a uniform magnetic field is suggested. The model can be conceived as a periodic lattice of resonators(curvilinear triangles)connected through ‘‘infinitely small’’ openings at the vertices of the triangles. The model Hamiltonian is obtained by means of operator extension theory in indefinite metric spaces. In the case of rational magnetic flux through an elementary cell of the lattice, the dispersion equation is found in an explicit form with the help of harmonic analysis on the magnetic translation group. It is proved, at least in the case of integer flux, that the spectrum of the model Hamiltonian consists of three parts: (1) Landau levels (they correspond to the classical orbits lying between antidots); (2) extended states that correspond to the classical propagation trajectories; and (3) bound states satisfying the dispersion equation; they correspond to the classical chaotic orbits rotating around single antidots. Among other things,...