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# Dynamics of an electron confined to a "hybrid plane" and interacting with a magnetic field

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*in*Reports on Mathematical Physics 67(2) · April 2011

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Abstract

We discuss spectral and resonance properties of a Hamiltonian describing
motion of an electron moving on a "hybrid surface" consisting on a halfline
attached by its endpoints to a plane under influence of a constant magnetic
field which interacts with its spin through a Rashba-type term.

- ... Variant forms of h were studied by many authors. For example, the special case α = 0 in various spatial dimensions can be found in[5][6][7][8][9]. A general case in three spatial dimensions is studied in[10,11]. ...... Variant forms of h were studied by many authors. For example, the special case α = 0 in various spatial dimensions can be found in [5][6][7][8][9]. A general case in three spatial dimensions is studied in [10,11]. ...... Variant forms of h were studied by many authors. For example, the special case α = 0 (but β ≥ 0) in various spatial dimensions can be found in [9][10][11][12][13]. A general case in three spatial dimensions is studied in [6,14]. ...We compute an explicit formula for the one-parameter unitary group of the single-particle Rashba spin-orbit coupled operator in dimension three. As an application, we derive the formula for the Green function for the two-particle operator, and then prove that the spin-dependent point-interaction is of class $\mathcal{H}_{-4}$. The latter is thus the example of a supersingular perturbation for which no self-adjoint operator can be constructed.
- ... From the theoretical point of view the hybrids are interesting because the conduction of quantum particles inside the hybrid is strictly related to the formation of quantum resonances, as in the case of quantum wires. Starting from some simplified models (see [16]) the development and refinement of spectral analysis techniques has allowed the possibility to consider more and more complicated structure or interactions (see [6], [17], [12], [10]). In this paper we apply results obtained in the case of models with point interactions and formation of resonances (see [7],[8], [9]). ...... In this section we analyze the spectral and scattering properties related to a manifold obtained by gluing the wedge with an half-line. This procedure has been widely studied in literature; the procedure allowing to paste varieties with different dimensionality is discussed in several papers (see [16, 12, 14, 15, 10]). The peculiarity of the present model lays in the fact that the sticking point is the vertex of the wedge, and along its boundary, outside the vertex itself, Dirichlet boundary condition hold. ...... For the construction of the hybrid we will follow a standard procedure for the construction of self-adjoint extensions (see e.g. [1], [18] for the general theory and [13], [10] for hybrids) starting from the Green function of the decoupled system. We define the boundary maps Γ k : H 2 (0, +∞) ⊕ D((∆ @BULLET ...Article
- Dec 2016
- PHYS LETT A

We study the spectrum, resonances and scattering matrix of a quantum hamiltonian on a "hybrid surface" consisting of a half-line attached by its endpoint to the vertex of a concave planar wedge. At the boundary of the wedge, outside the vertex, Dirichlet boundary are imposed. The system is tunable by varying the measure of the angle at the vertex. - ... A modern and comprehensive review of various operator techniques is given in [15]. In one and two spatial dimensions, the spectral properties of self-adjoint extensions of S, for α, β ě 0, were studied in [14,16,17,20,27,31]. A mean field interpretation, which is commonly accepted in physics literature, can be found in [29,32,34,35]. ...... has been derived. In contrast, the analogous result in two spatial dimensions possesses a closed form [14,20,27]. The absolute convergence of hypergeometric series causes the limitation on the parameters α, β ě 0. The result is that the infimum of the continuous spectrum r´Σ, 8q must satisfy the condition´Σcondition´Σ ě ´1. ...The present study is the first such attempt to examine rigorously and comprehensively the spectral properties of a three-dimensional ultracold atom when both the spin-orbit interaction and the Zeeman field are taken into account. The model operator is the Rashba spin-orbit coupled operator in dimension three. The self-adjoint extensions are constructed using the theory of singular perturbations, where regularized rank two perturbations describe spin-dependent contact interactions. The spectrum of self-adjoint extensions is investigated in detail laying emphasis on the effects due to spin-orbit coupling. When the spin-orbit-coupling strength is small enough, the asymptotics of eigenvalues is obtained. The conditions for the existence of eigenvalues above the threshold are discussed in particular.
- ... The special case when α = 0 has been discussed in [3]. A two-dimensional equivalent of (1.3), with p = (p 1 , p 2 ) ∈ R 2 and x ∈ R 2 , has been computed in456. In fact, the derivation of the Green function in dimension two does not require an explicit calculation of (1.3), for one explores the fact that the square of the spin-orbit term in the Hamiltonian is just the two-dimensional Laplace operator. ...... The special case when α = 0 has been discussed in [3]. A two dimensional equivalent of (1.3), with p = (p 1 , p 2 ) ∈ R 2 and x ∈ R 2 , has been computed in [4,5,6]. In fact, the derivation of the Green's function in dimension two does not require an explicit calculation of (1.3), for one explores the fact that the square of spin-orbit term in the Hamiltonian is just the two-dimensional Laplace operator. ...In Rashba-Dresselhaus spin-orbit coupled systems, the calculation of Green's function requires the knowledge of the inverse Fourier transform of rational function $P(p)/Q(p)$, where $P(p)$ takes the values $1$ and $p^{2}$, and where \[ Q(p)=(p^{2}-\zeta)^{2}- \alpha^{2}(p_{1}^{2}+p_{2}^{2})-\beta^{2} \] with suitable parameters $\alpha$, $\beta\geq0$, $\zeta\in\mathbb{C}$. While a two-dimensional problem, with $p=(p_{1},p_{2})$, has been recently solved [J. Br\"{u}ning et al, J. Phys. A: Math. Theor. 40 (2007)], its three-dimensional analogue, with $p=(p_{1},p_{2},p_{3})$, remains open. In this paper, a hypergeometric series expansion for the triple integral is provided. Convergence of the series dependent on the parameters is studied in detail.
- ... Linear point interactions arise as a particular, but relevant, application of the more general theory of self-adjoint extension of symmetric operators; a theory that has gained new popularity in recent years also for the application to the study of evolution equations in non-standard domains, such as quantum graphs (see e.g., [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]) and quantum hybrids (see e.g., [27][28][29][30][31]). ...ArticleFull-text available
- Apr 2018

We present an introduction to the nonlinear Schr\"odinger equation (NLSE) with concentrated nonlinearities in $\mathbb{R}^2$. Precisely, taking a cue from the linear problem, we sketch the main challenges and the typical difficulties that arise in the two-dimensional case, and mention some recent results obtained by the authors on local and global well-posedness. - ... The mathematical model is based on the theory of self-adjoint extensions of symmetric operators (see e.g.[12][13][14]and references in[15]). Solvable models of such type for analogous problems were considered in many papers (see e.g.[16][17][18][19][20][21][22]). Completeness of resonance states for the simplest quantum graph—line with segment attached—was proved in[23]. ...3D quantum dot with two 1D wires attached is considered. Scattering matrix is constructed in the framework of Lax—Phillips approach. The completeness of resonant states is proved using the factorization criterion for the characteristic function of Sz.-Nagy functional model.
- ArticleFull-text available
- Apr 2018

- Book
- Jan 2013

This volume presents self-contained survey articles on modern research areas written by experts in their fields. The topics are located at the interface of spectral theory, theory of partial differential operators, stochastic analysis, and mathematical physics. The articles are accessible to graduate students and researches from other fields of mathematics or physics while also being of value to experts, as they report on the state of the art in the respective fields. - Spectral properties of periodic one-dimensional array of nanorings in a magnetic field are investigated. Two types of the superlattice are considered. In the first one, rings are connected by short one-dimensional wires while in the second one rings have immediate contacts between each other. The dependence of the electron energy on the quasimomentum is obtained from the Schrodinger equation for the Bloch wave function. We have found an interesting feature of the system, namely, presence of discrete energy levels in the spectrum. The levels can be located in the gaps or in the bands depending on parameters of the system. The levels correspond to bound states and electrons occupying these levels are located on individual rings or couples of neighbouring rings and does not contribute to the charge transport. The wave function for the bound states corresponding to the discrete levels is obtained. Modification of electron energy spectrum with variation of system parameters is discussed.
- Article
- May 2012

Resonance and decay phenomena are ubiquitous in the quantum world. To understand them in their complexity it is useful to study solvable models in a wide sense, that is, systems which can be treated by analytical means. The present review offers a survey of such models starting the classical Friedrichs result and carrying further to recent developments in the theory of quantum graphs. Our attention concentrates on dynamical mechanism underlying resonance effects and at time evolution of the related unstable systems.

- The Landau Hamiltonian governing the behavior of a quantum particle in dimension 2 in a constant magnetic field is perturbed by a compactly supported magnetic field and a similar electric field. We describe how the spectral subspaces change and how the Landau levels split under this perturbation.
- Article
- Jan 2007

An algebraic algorithm is developed for computation of invariants ('generalized Casimir operators') of general Lie algebras over the real or complex number field. Its main tools are the Cartan's method of moving frames and the knowledge of the group of inner automorphisms of each Lie algebra. Unlike the first application of the algorithm in Boyko et al (2006 J. Phys. A: Math. Gen. 39 5749 (Preprint math-ph/0602046)), which deals with low-dimensional Lie algebras, here the effectiveness of the algorithm is demonstrated by its application to computation of invariants of solvable Lie algebras of general dimension n < ∞ restricted only by a required structure of the nilradical. Specifically, invariants are calculated here for families of real/complex solvable Lie algebras. These families contain, with only a few exceptions, all the solvable Lie algebras of specific dimensions, for whom the invariants are found in the literature. - Article
- Jul 1988
- J Math Sci

For nonadditive finite-dimensional perturbations of a Hermitian operator, with the aid of M. G. Krein's formula for generalized resolvents, one derives a representation of all scattering suboperators, parametrized by Hermitian matrices. On the basis of this representation, one obtains explicit expressions for a series of new, exactly solvable quantum models with null-range potential. One establishes a connection between the obtained parametrizations with phenomenological S matrices and the corresponding Wigner R functions. - In this paper, the free motion of a particle on a manifold that consists of a one‐dimensional and a two‐dimensional part connected in one point is discussed. The class of admissible Hamiltonians is found using the theory of self‐adjoint extensions. Particular attention is paid to those Hamiltonians that allow the particle to pass through the point singularity; the reflection coefficient and other quantities characterizing scattering on the connection point are calculated. A possible application is also discussed.
- Article
- Sep 2001
- J MATH PHYS

We study transport in quantum systems consisting of a finite array of N identical single-channel scatterers. A general expression of the S matrix in terms of the individual-element data obtained recently for potential scattering is rederived in this wider context. It shows in particular how the band spectrum of the infinite periodic system arises in the limit N→∞. We illustrate the result on two kinds of examples. The first are serial graphs obtained by chaining loops or T-junctions. Another example concerns geometric scatterers where the individual element consists of a surface with a pair of leads; we show that apart from the resonances coming from the decoupled-surface eigenvalues, such scatterers exhibit the high-energy behavior typical for the δ′ interaction for the physically interesting couplings. © 2001 American Institute of Physics. - ArticleFull-text available
- Jul 1988

For nonadditive finite-dimensional perturbations of a Hermitian operator, with the aid of M. G. Krein's formula for generalized resolvents, one derives a representation of all scattering suboperators, parametrized by Hermitian matrices. On the basis of this representation, one obtains explicit expressions for a series of new, exactly solvable quantum models with null-range potential. One establishes a connection between the obtained parametrizations with phenomenological S matrices and the corresponding Wigner R functions. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 7–23, 1986. - Article
- Sep 2009

We discuss approximations of vertex couplings of quantum graphs using families of thin branched manifolds. We show that if a Neumann-type Laplacian on such manifolds is amended by suitable potentials, the resulting Schrödinger operators can approximate non-trivial vertex couplings. The latter include not only the δ-couplings but also those with wavefunctions discontinuous at the vertex. We work out the example of the symmetric δ'-couplings and make a conjecture that the same method can be applied to all couplings invariant with respect to the time reversal. We conclude with a result that certain vertex couplings cannot be approximated by a pure Laplacian. - Article
- Jan 1999
- J Phys Math Gen

We study the scattering problem on a graph consisting of a line with a finite-length appendix. The two parts are coupled through boundary conditions depending on three parameters; the motion on the line is free while the appendix supports a potential. The appendix bound states give rise to a ladder of resonances; we construct the resolvent and solve the corresponding pole condition for a weak coupling. In general, the condition only admits an analytic solution in particular cases. We find the pole positions numerically for a linear potential and show that the poles eventually return to the real axis when the coupling strength increases. - Article
- May 2002
- J Phys Math Gen

The conductance of a quantum sphere with two one-dimensional wires attached to it is investigated. An explicit form for the conductance as a function of the chemical potential is found from first principles. The form and positions of the resonance maxima on the plot of the conductance are studied. - Article
- Feb 1997
- PHYS LETT A

We propose a model for scattering in a flat resonator with a thin antenna. The results are applied to rectangular microwave cavities. We compute the resonance spacing distribution and show that it agrees well with experimental data provided the antenna radius is much smaller than wavelengths of the resonance wavefunctions. - Article
- Dec 2008

We discuss approximations of vertex couplings of quantum graphs using families of thin branched manifolds. We show that if a Neumann type Laplacian on such manifolds is amended by suitable potentials, the resulting Schr\"odinger operators can approximate non-trivial vertex couplings. The latter include not only the delta-couplings but also those with wavefunctions discontinuous at the vertex. We work out the example of the symmetric delta'-couplings and conjecture that the same method can be applied to all couplings invariant with respect to the time reversal. - Article
- Jan 2003
- J Phys Math Gen

We study a free quantum motion on periodically structured manifolds composed of elementary two-dimensional "cells" connected either by linear segments or through points where the two cells touch. The general theory is illustrated with numerous examples in which the elementary components are spherical surfaces arranged into chains in a straight or zigzag way, or two-dimensional square-lattice "carpets". We show that the spectra of such systems have an infinite number of gaps and that the latter dominate the spectrum at high energies. 1 - Article
- Jun 2002

In the paper [25]... In this paper we consider the quantum mechanical scattering in a hedgehog-shaped space which is constructed by gluing a finite number of half-lines to distinct points of a compact Riemannian manifold of dimension less than four. The Hamiltonian of a quantum particle in such a system coincides with a Schrödinger operator on the punctured manifold (the points of gluing are removed) and with the free Schrödinger operator on each half-line. At the gluing points, some boundary conditions are imposed. In particular, the Schröodinger operator in a magnetic field is included in our scheme. The approach we use is based on the Krein resolvent formula from operator extension theory [50], therefore in Sec. 1 we give a very brief sketch of results needed from this theory. Sec. 2 is devoted to the construction of Schrödinger operators on the hedgehog-shaped space; we use the theory of boundary value spaces [35] to describe all possible kinds of boundary conditions defining the Schrödinger operators. We distinguish among them operators of "Dirichlet" and of "Neumann" type. It is worth noting that the results of Sec. 2 are valid for all Riemannian manifolds of dimension less than four, not only for the compact ones. In principle, the definition of the Schrödinger operator on a hedgehog-shaped space may be given in the framework of pseudo-differential operator theory on such a space [66], but our approach is more convenient for investigating the scattering parameters and connected with the approach to spectral problems for point perturbations on Riemannian manifolds [8], [9]... - In this article we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with $n$ open ends and where we define the Hamiltonian to be (minus) the Laplace operator with general boundary conditions at the vertices. This results in a scattering theory with $n$ channels. The corresponding on-shell S-matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy $E>0$ is explicitly given in terms of the boundary conditions and the lengths of the internal lines. It is shown to be unitary, which may be viewed as the quantum version of Kirchhoff's law. We exhibit covariance and symmetry properties. It is symmetric if the boundary conditions are real. Also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory. Finally we provide a composition rule by which the on-shell S-matrix of a graph is factorizable in terms of the S-matrices of its subgraphs. All proofs only use known facts from the theory of self-adjoint extensions, standard linear algebra, complex function theory and elementary arguments from the theory of Hermitean symplectic forms.
- Article
- Dec 2006
- J PHYS A-MATH THEOR

We give a variational proof of the existence of infinitely many bound states below the continuous spectrum for some weak perturbations of a class of spin-orbit Hamiltonians including the Rashba and Dresselhaus Hamiltonians. - For a scattering system {AΘ, A0} consisting of self-adjoint extensions AΘ and A0 of a symmetric operator A with finite deficiency indices, the scattering matrix {SΘ(λ)} and a spectral shift function ξΘ are calculated in terms of the Weyl function associated with a boundary triplet for A*, and a simple proof of the Krein–Birman formula is given. The results are applied to singular Sturm–Liouville operators with scalar and matrix potentials, to Dirac operators and to Schrödinger operators with point interactions.
- We prove an analog of Krein's resolvent formula expressing the resolvents of self-adjoint extensions in terms of boundary conditions. Applications to quantum graphs and systems with point interactions are discussed.
- Article
- May 2002
- J MATH PHYS

The quantum-mechanical scattering on a compact Riemannian manifold with semi-axes attached to it (hedgehog-shaped manifold) is considered. The complete description of the spectral structure of Schroedinger operators on such a manifold is done, the proof of existence and uniqueness of scattering states is presented, an explicit form for the scattering matrix is obtained and unitary nature of this matrix is proven. It is shown that the positive part of the spectrum of the Schroedinger operator on the initial compact manifold as well as the spectrum of a point perturbation of such an operator may be recovered from the scattering amplitude for one attached half-line. Moreover, the positive part of the spectrum of the initial Schroedinger operator is fully determined by the conductance properties of an "electronic device" consisting of the initial manifold and two "wires" attached to it. - In this paper we attempt to reconstruct one of the last projects of Volodya Geyler which remained unfinished. We study motion of a quantum particle in the plane to which a halfline lead is attached assuming that the particle has spin $\frac12$ and the plane component of the Hamiltonian contains a spin-orbit interaction of either Rashba or Dresselhaus type. We construct the class of admissible Hamiltonians and derive an explicit expression for the Green function applying it to the scattering in such a system. Comment: LaTeX, 9 pages; in memoriam Vladimir A. Geyler (1943-2007)
- We derive explicit expressions for Green functions and some related characteristics of the Rashba and Dresselhaus Hamiltonians with a uniform magnetic field.