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Feb 2012

Volume 53, Issue 2, Articles (02xxxx)

Issue Cover Spotlight Figure

J. Math. Phys. 53, 023508 (2012); http://dx.doi.org/10.1063/1.3679686 (45 pages)

Samuel Friot and David Greynat
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back to top Quantum Mechanics (General and Nonrelativistic)

Convergence of expansions in Schrödinger and Dirac eigenfunctions, with an application to the R-matrix theory

Julia Stasińska

J. Math. Phys. 53, 022101 (2012); http://dx.doi.org/10.1063/1.3679763 (12 pages)

Online Publication Date: 2 February 2012

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Expansion of a wave function in a basis of eigenfunctions of a differential eigenvalue problem lies at the heart of the R-matrix methods for both the Schrödinger and Dirac particles. A central issue that should be carefully analyzed when functional series are applied is their convergence. In the present paper, we study the properties of the eigenfunction expansions appearing in nonrelativistic and relativistic R-matrix theories. In particular, we confirm the findings of Rosenthal [J. Phys. G 13, 491 (1987)] and Szmytkowski and Hinze [J. Phys. B 29, 761 (1996); Szmytkowski and Hinze J. Phys. A 29, 6125 (1996)] that in the most popular formulation of the R-matrix theory for Dirac particles, the functional series fails to converge to a limit claimed by other authors.
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03.65.Ge Solutions of wave equations: bound states
02.10.Ud Linear algebra

Delay time in quaternionic quantum mechanics

Stefano De Leo and Gisele Ducati

J. Math. Phys. 53, 022102 (2012); http://dx.doi.org/10.1063/1.3684747 (8 pages)

Online Publication Date: 13 February 2012

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In looking for quaternionic violations of quantum mechanics, we discuss the delay time for pure quaternionic potentials. Our study shows the energy region which amplifies the difference between quaternionic and complex quantum mechanics.
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03.65.-w Quantum mechanics

Generalized five-dimensional Kepler system, Yang-Coulomb monopole, and Hurwitz transformation

Ian Marquette

J. Math. Phys. 53, 022103 (2012); http://dx.doi.org/10.1063/1.3684955 (12 pages)

Online Publication Date: 15 February 2012

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The 5D Kepler system possesses many interesting properties. This system is superintegrable and also with a su(2) non-Abelian monopole interaction (Yang-Coulomb monopole). This system is also related to an 8D isotropic harmonic oscillator by a Hurwitz transformation. We introduce a new superintegrable Hamiltonian that consists in a 5D Kepler system with new terms of Smorodinsky-Winternitz type. We obtain the integrals of motion of this system. They generate a quadratic algebra with structure constants involving the Casimir operator of a so(4) Lie algebra. We also show that this system remains superintegrable with a su(2) non-Abelian monopole (generalized Yang-Coulomb monopole). We study this system using parabolic coordinates and obtain from Hurwitz transformation its dual that is an 8D singular oscillator. This 8D singular oscillator is also a new superintegrable system and multiseparable. We obtained its quadratic algebra that involves two Casimir operators of so(4) Lie algebras. This correspondence is used to obtain algebraically the energy spectrum of the generalized Yang-Coulomb monopole.
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11.15.-q Gauge field theories
11.30.Ly Other internal and higher symmetries
12.20.Ds Specific calculations
02.20.Sv Lie algebras of Lie groups

Dirac equation for generalized Pöschl-Teller scalar and vector potentials and a Coulomb tensor interaction by Nikiforov-Uvarov method

H. Hassanabadi, E. Maghsoodi, S. Zarrinkamar, and H. Rahimov

J. Math. Phys. 53, 022104 (2012); http://dx.doi.org/10.1063/1.3683078 (11 pages)

Online Publication Date: 15 February 2012

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Approximate analytical solutions of spin and pseudospin symmetry limits of Dirac equation are reported for the generalized Pöschl-Teller scalar and vector potentials and a Coulomb tensor interaction by Nikiforov-Uvarov method. On the contrary to the cumbersome numerical procedures, the analytical approach followed here can be followed even by the undergraduate students.
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03.65.Pm Relativistic wave equations
02.10.Ud Linear algebra

Mean-field quantum dynamics with magnetic fields

Jonas Lührmann

J. Math. Phys. 53, 022105 (2012); http://dx.doi.org/10.1063/1.3687024 (19 pages)

Online Publication Date: 24 February 2012

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We consider a system of N bosons in three dimensions interacting through a mean-field Coulomb potential in an external magnetic field. For initially factorized states we show that the one-particle density matrix associated with the solution of the N-body Schrödinger equation converges to the projection onto the solution of the magnetic Hartree equation in trace norm and in energy as N → ∞. Estimates on the rate of convergence are provided.
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03.65.Ge Solutions of wave equations: bound states
02.30.Hq Ordinary differential equations
05.30.Jp Boson systems
02.60.Lj Ordinary and partial differential equations; boundary value problems
back to top Quantum Information and Computation

Local quasi hidden variable modelling and violations of Bell-type inequalities by a multipartite quantum state

Elena R. Loubenets

J. Math. Phys. 53, 022201 (2012); http://dx.doi.org/10.1063/1.3681905 (30 pages)

Online Publication Date: 15 February 2012

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We introduce for a general correlation scenario a new simulation model, a local quasi hidden variable (LqHV) model, where locality and the measure-theoretic construction inherent to a local hidden variable (LHV) model are preserved but positivity of a simulation measure is dropped. We specify a necessary and sufficient condition for LqHV modelling and, based on this, prove that every quantum correlation scenario admits a LqHV simulation. Via the LqHV approach, we construct analogs of Bell-type inequalities for an N-partite quantum state and find a new analytical upper bound on the maximal violation by an N-partite quantum state of S1 × ⋯ × SN-setting Bell-type inequalities – either on correlation functions or on joint probabilities and for outcomes of an arbitrary spectral type, discrete or continuous. This general analytical upper bound is expressed in terms of the new state dilation characteristics introduced in the present paper and not only traces quantum states admitting an S1 × ⋯ × SN-setting LHV description but also leads to the new exact numerical upper estimates on the maximal Bell violations for concrete N-partite quantum states used in quantum information processing and for an arbitrary N-partite quantum state. We, in particular, prove that violation by an N-partite quantum state of an arbitrary Bell-type inequality (either on correlation functions or on joint probabilities) for S settings per site cannot exceed (2S − 1)N − 1 even in case of an infinite dimensional quantum state and infinitely many outcomes.
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03.65.Ta Foundations of quantum mechanics; measurement theory
03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)
03.67.-a Quantum information
02.50.Cw Probability theory

Rank reduction for the local consistency problem

Jianxin Chen, Zhengfeng Ji, Alexander Klyachko, David W. Kribs, and Bei Zeng

J. Math. Phys. 53, 022202 (2012); http://dx.doi.org/10.1063/1.3685644 (8 pages)

Online Publication Date: 21 February 2012

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We address the problem of how simple a solution can be for a given quantum local consistency instance. More specifically, we investigate how small the rank of the global density operator can be if the local constraints are known to be compatible. We prove that any compatible local density operators can be satisfied by a low rank global density operator. Then we study both fermionic and bosonic versions of the N-representability problem as applications. After applying the channel-state duality, we prove that any compatible local channels can be obtained through a global quantum channel with small Kraus rank.
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03.65.Ta Foundations of quantum mechanics; measurement theory
05.30.Jp Boson systems
05.30.Fk Fermion systems and electron gas
back to top Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

Solitonic solutions of Faddeev model

Chang-Guang Shi and Minoru Hirayama

J. Math. Phys. 53, 022301 (2012); http://dx.doi.org/10.1063/1.3682246 (8 pages)

Online Publication Date: 7 February 2012

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An application of the equation proposed by the present authors, which is equivalent to the static field equation of the Faddeev model, is discussed. Under some assumptions on the space and on the form of the solution, the field equation is reduced to a nonlinear ordinary differential equation of second order. By solving this equation numerically, some solitonic solutions are obtained. It is discussed that the product of two integers specifying solutions may be identified with the Hopf topological invariant.
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11.80.Jy Many-body scattering and Faddeev equation
02.40.Pc General topology
02.30.Hq Ordinary differential equations
11.10.Lm Nonlinear or nonlocal theories and models

On flavor symmetry in lattice quantum chromodynamics

El Hassan Saidi

J. Math. Phys. 53, 022302 (2012); http://dx.doi.org/10.1063/1.3682640 (26 pages)

Online Publication Date: 9 February 2012

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Using a well established method to engineer non-abelian symmetries in superstring compactifications, we study the link between the point splitting method of Creutz et al. [PoS: Lattice 2010, 078 (2010) and Creutz et al. JHEP 041, 1012 (2010)] for implementing flavor symmetry in lattice QCD; and singularity theory in complex algebraic geometry. We show amongst others that Creutz flavors for naive fermions are intimately related with toric singularities of a class of complex Kahler manifolds that are explicitly built here. In the case of naive fermions of QCD2N, Creutz flavors are shown to live at the poles of real 2-spheres and carry quantum charges of the fundamental of [SU(2)]2N. We show moreover that the two Creutz flavors in Karsten-Wilczek model, with Dirac operator in reciprocal space of the form iγ1F1+iγ2F2+iγ3F3+mathγ4F4, are related with the small resolution of conifold singularity that live at sin α = 0. Other related features are also studied.
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12.38.Aw General properties of QCD (dynamics, confinement, etc.)
11.30.Ly Other internal and higher symmetries
11.25.-w Strings and branes
12.39.-x Phenomenological quark models
12.38.Gc Lattice QCD calculations
11.10.Cd Axiomatic approach

Noncommutative deformation of spinor zero mode and Atiyah-Drinfeld-Hitchin-Manin construction

Yoshiaki Maeda and Akifumi Sako

J. Math. Phys. 53, 022303 (2012); http://dx.doi.org/10.1063/1.3679398 (24 pages)

Online Publication Date: 10 February 2012

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A method to construct noncommutative instantons as deformations from commutative instantons was provided by Maeda and Sako [J. Geom. Phys. 58, 1784 (2008)]10.1016/j.geomphys.2008.08.006. Using this noncommutative deformed instanton, we investigate the spinor zero modes of the Dirac operator in a noncommutative instanton background on noncommutative math4, and we modify the index of the Dirac operator on the noncommutative space slightly and show that the number of the zero mode of the Dirac operator is preserved under the noncommutative deformation. We prove the existence of the Green's function associated with instantons on noncommutative math4, as a smooth deformation of the commutative case. The feature of the zero modes of the Dirac operator and the Green's function derives noncommutative ADHM (Atiyah-Drinfeld-Hitchin-Manin) equations which coincide with the ones introduced by Nekrasov and Schwarz. We show a one-to-one correspondence between the instantons on noncommutative math4 and ADHM data. An example of a noncommutative instanton and a spinor zero mode are also given.
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11.10.Nx Noncommutative field theory
02.30.Sa Functional analysis

The coincidence limit of the graviton propagator in de Donder gauge on de Sitter background

E. O. Kahya, S. P. Miao, and R. P. Woodard

J. Math. Phys. 53, 022304 (2012); http://dx.doi.org/10.1063/1.3681886 (28 pages)

Online Publication Date: 10 February 2012

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We explicitly work out the de Sitter breaking contributions to the recent solution for the de Donder gauge graviton propagator on de Sitter. We also provide explicit power series expansions for the two structure functions, which are suitable for implementing dimensional regularization. And we evaluate the coincidence limit of the propagator.
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04.60.-m Quantum gravity
14.70.Kv Gravitons

Regularized path integrals and anomalies: U(1) chiral gauge theory

Christoph Kopper and Benjamin Lévêque

J. Math. Phys. 53, 022305 (2012); http://dx.doi.org/10.1063/1.3686000 (24 pages)

Online Publication Date: 23 February 2012

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We analyze the origin of the Adler–Bell–Jackiw anomaly of chiral U(1) gauge theory within the framework of regularized path integrals. Momentum or position space regulators allow for mathematically well-defined path integrals but violate local gauge symmetry. It is known how (nonanomalous) gauge symmetry can be recovered in the renormalized theory in this case [Kopper, C. and Müller, V. F., “Renormalization of spontaneously broken SU(2) Yang-Mills theory with flow equations,” Rev. Math. Phys. 21, 781 (2009)]10.1142/S0129055X0900375X. Here we analyze U(1) chiral gauge theory to show how the appearance of anomalies manifests itself in such a context. We show that the three-photon amplitude leads to a violation of the Slavnov-Taylor identities which cannot be restored on taking the UV limit in the renormalized theory. We point out that this fact is related to the nonanalyticity of this amplitude in the infrared region.
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11.15.-q Gauge field theories
11.30.Rd Chiral symmetries
11.10.Gh Renormalization
back to top General Relativity and Gravitation

Quantum deformation of two four-dimensional spin foam models

Winston J. Fairbairn and Catherine Meusburger

J. Math. Phys. 53, 022501 (2012); http://dx.doi.org/10.1063/1.3675898 (37 pages)

Online Publication Date: 2 February 2012

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We construct the q-deformed version of two four-dimensional spin foam models, the Euclidean and Lorentzian versions of the Engle, Pereira, Rovelli and Livine (EPRL) model. The q-deformed models are based on the representation theory of two copies of Uq(math(2)) at a root of unity and on the quantum Lorentz group with a real deformation parameter. For both models, we give a definition of the quantum EPRL intertwiners, study their convergence and braiding properties, and construct an amplitude for the four-simplexes. We find that both of the resulting models are convergent.
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03.65.Fd Algebraic methods
03.65.Ta Foundations of quantum mechanics; measurement theory
02.20.Sv Lie algebras of Lie groups

Spacelike hypersurfaces with negative total energy in de Sitter spacetime

Zhuobin Liang and Xiao Zhang

J. Math. Phys. 53, 022502 (2012); http://dx.doi.org/10.1063/1.3682242 (10 pages)

Online Publication Date: 3 February 2012

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De Sitter spacetime can be separated into two parts along two kinds of hypersurfaces and the half-de Sitter spacetimes are covered by the planar and hyperbolic coordinates, respectively. Two positive energy theorems were proved previously for certain P-asymptotically de Sitter and H-asymptotically de Sitter initial data sets by the second author and collaborators. These initial data sets are asymptotic to time slices of the two kinds of half-de Sitter spacetimes, respectively, and their mean curvatures are bounded from above by certain constants. While the mean curvatures violate these conditions, the spacelike hypersurfaces with negative total energy in the two kinds of half-de Sitter spacetimes are constructed in this short paper.
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04.20.Gz Spacetime topology, causal structure, spinor structure

Asymptotic stability of vacuum twisting type II metrics

Włodzimierz Natorf

J. Math. Phys. 53, 022503 (2012); http://dx.doi.org/10.1063/1.3682273 (4 pages)

Online Publication Date: 7 February 2012

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We generalize the result of Lukács et al. on asymptotic stability of the Schwarzschild metric with respect to perturbations in the Robinson-Trautman class of metrics to the case of Petrov type II twisting metrics, under the condition of asymptotic flatness at future null infinity. The Bondi energy is used as the Lyapunov functional and we prove that the “final state” of such metrics is the Kerr metric.
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04.70.-s Physics of black holes
04.20.Gz Spacetime topology, causal structure, spinor structure
back to top Dynamical Systems

The holonomy group at infinity of the Painlevé VI equation

Bassem Ben Hamed, Lubomir Gavrilov, and Martine Klughertz

J. Math. Phys. 53, 022701 (2012); http://dx.doi.org/10.1063/1.3681897 (15 pages)

Online Publication Date: 3 February 2012

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We prove that the holonomy group at infinity of the Painlevé VI equation is virtually commutative.
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02.30.Hq Ordinary differential equations

Total destruction of invariant tori for the generalized Frenkel–Kontorova model

Xifeng Su and Lin Wang

J. Math. Phys. 53, 022702 (2012); http://dx.doi.org/10.1063/1.3683233 (7 pages)

Online Publication Date: 13 February 2012

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We consider generalized Frenkel–Kontorova models on higher dimensional lattices. We show that the invariant tori which are parameterized by continuous hull functions can be destroyed by small perturbations in the Cr topology with r < 1.
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
02.40.Pc General topology

Dynamics of electrostatic microelectromechanical systems actuators

Yisong Yang, Ruifeng Zhang, and Le Zhao

J. Math. Phys. 53, 022703 (2012); http://dx.doi.org/10.1063/1.3684748 (13 pages)

Online Publication Date: 13 February 2012

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Electrostatic actuators are simple but important switching devices for microelectromechanical systems applications. Due to the difficulties associated with the electrostatic nonlinearity, precise mathematical description is often hard to obtain for the dynamics of these actuators. Here we present two sharp theorems concerning the dynamics of an undamped electrostatic actuator with one-degree of freedom, subject to linear and nonlinear elastic forces, respectively. We prove that both situations are characterized by the onset of one-stagnation-point periodic response below a well-defined pull-in voltage and a finite-time touch-down or collapse of the actuator above this pull-in voltage. In the linear-force situation, the stagnation level, pull-in voltage, and pull-in coordinate of the movable electrode may all be determined explicitly, following the recent work of Leus and Elata based on numerics. Furthermore, in the nonlinear-force situation, the stagnation level, pull-in voltage, and pull-in coordinate may be described completely in terms of the electrostatic and mechanical parameters of the model so that they approach those in the linear-force situation monotonically in the zero nonlinear-force limit.
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07.10.Cm Micromechanical devices and systems
84.32.Dd Connectors, relays, and switches
85.85.+j Micro- and nano-electromechanical systems (MEMS/NEMS) and devices
back to top Classical Mechanics and Classical Fields

Classical group field theory

Joseph Ben Geloun

J. Math. Phys. 53, 022901 (2012); http://dx.doi.org/10.1063/1.3682651 (31 pages)

Online Publication Date: 9 February 2012

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The ordinary formalism for classical field theory is applied to dynamical group field theories. Focusing first on a local group field theory over one copy of SU(2) and then, on more involved nonlocal theories (colored and noncolored) defined over a tensor product of the same group, we address the issue of translation and dilatation symmetries and the corresponding Noether theorem. The energy momentum tensor and dilatation current are derived and their properties identified for each case.
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03.50.-z Classical field theories
02.20.-a Group theory
back to top Statistical Physics

Proof of rounding by quenched disorder of first order transitions in low-dimensional quantum systems

Michael Aizenman, Rafael L. Greenblatt, and Joel L. Lebowitz

J. Math. Phys. 53, 023301 (2012); http://dx.doi.org/10.1063/1.3679069 (20 pages)

Online Publication Date: 1 February 2012

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We prove that for quantum lattice systems in d ⩽ 2 dimensions the addition of quenched disorder rounds any first order phase transition in the corresponding conjugate order parameter, both at positive temperatures and at T = 0. For systems with continuous symmetry the statement extends up to d ⩽ 4 dimensions. This establishes for quantum systems the existence of the Imry–Ma phenomenon which for classical systems was proven by Aizenman and Wehr. The extension of the proof to quantum systems is achieved by carrying out the analysis at the level of thermodynamic quantities rather than equilibrium states.
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03.65.Ta Foundations of quantum mechanics; measurement theory
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
05.70.Fh Phase transitions: general studies

A dimension scale-invariant probabilistic model based on Leibniz-like pyramids

A. Rodríguez and C. Tsallis

J. Math. Phys. 53, 023302 (2012); http://dx.doi.org/10.1063/1.3688312 (27 pages)

Online Publication Date: 27 February 2012

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We introduce a family of dimension scale-invariant Leibniz-like pyramids and (d + 1)-dimensional hyperpyramids (d = 1, 2, 3, …), with d = 1 corresponding to triangles, d = 2 to (tetrahedral) pyramids, and so on. For all values of d, they are characterized by a parameter ν > 0, whose value determines the degree of correlation between N (d + 1)-valued random variables (d = 1 corresponds to binary variables, d = 2 to ternary variables, and so on). There are (d + 1)N different events, and the limit ν → ∞ corresponds to independent random variables, in which case each event has a probability 1/(d + 1)N to occur. The sums of these N  (d + 1)-valued random variables correspond to a d-dimensional probabilistic model and generalize a recently proposed one-dimensional (d = 1) model having q −Gaussians (with q = (ν − 2)/(ν − 1) for ν ∈ [1, ∞)) as N → ∞ limit probability distributions for the sum of the N binary variables [A. Rodríguez, V. Schwammle, and C. Tsallis, J. Stat. Mech.: Theory Exp. 2008, P09006; R. Hanel, S. Thurner, and C. Tsallis, Eur. Phys. J. B 72, 263 (2009)]. In the ν → ∞ limit the d-dimensional multinomial distribution is recovered for the sums, which approach a d-dimensional Gaussian distribution for N → ∞. For any ν, the conditional distributions of the d-dimensional model are shown to yield the corresponding joint distribution of the (d−1)-dimensional model with the same ν. For the d = 2 case, we study the joint probability distribution and identify two classes of marginal distributions, one of them being asymmetric and dimension scale-invariant, while the other one is symmetric and only asymptotically dimension scale-invariant. The present probabilistic model is proposed as a testing ground for a deeper understanding of the necessary and sufficient conditions for having q-Gaussian attractors in the N → ∞ limit, the ultimate goal being a neat mathematical view of the causes clarifying the ubiquitous emergence of q-statistics verified in many natural, artificial, and social systems.
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02.50.Cw Probability theory

Asymptotic integral kernel for ensembles of random normal matrices with radial potentials

Alexei M. Veneziani, Tiago Pereira, and Domingos H. U. Marchetti

J. Math. Phys. 53, 023303 (2012); http://dx.doi.org/10.1063/1.3688293 (21 pages)

Online Publication Date: 28 February 2012

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The method of steepest descent is used to study the integral kernel of a family of normal random matrix ensembles with eigenvalue distribution PN(z1,⋯,zN) = ZN−1e−N∑i = 1NVα(zi)∏1 ≤ i<j ≤ N|zi−zj|2,where Vα(z) = |z|α, z∈math and α ∈ ]0, ∞[. Asymptotic formulas with error estimate on sectors are obtained. A corollary of these expansions is a scaling limit for the n-point function in terms of the integral kernel for the classical Segal–Bargmann space.
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.50.-r Probability theory, stochastic processes, and statistics
02.60.-x Numerical approximation and analysis
02.10.Yn Matrix theory
02.10.Ud Linear algebra
back to top Methods of Mathematical Physics

Local trace formulae and scaling asymptotics for general quantized Hamiltonian flows

Roberto Paoletti

J. Math. Phys. 53, 023501 (2012); http://dx.doi.org/10.1063/1.3679660 (22 pages)

Online Publication Date: 1 February 2012

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A class of trace formulae in the Toeplitz quantization of Hamiltonian flows on compact symplectic manifolds is given a local intepretation in terms of certain scaling asymptotics near a symplectic fixed locus; these are reminiscent of the near diagonal asymptotics of equivariant Szegö kernels. In spectral theory, the distributional trace of a positive elliptic operator encapsulates information on the asymptotic distribution of its eigenvalues; a trace formula relates the singularities of the trace to Poincaré data of an appropriate Hamiltonian flow. This leads to the asymptotics of certain smoothing kernels, related to the Fourier transform of the trace of a wave operator. In Toeplitz quantization, we consider analogues of these, with the wave operator replaced by a quantized Hamiltonian flow, possibly composed with a family of zeroth order Toeplitz operators. We study the concentration of these smoothing kernels near the fixed loci of the linearized dynamics, and specifically how the Poincaré map controls their scaling asymptotics. This generalizes previous results on holomorphic flows.
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02.30.Uu Integral transforms
02.10.Ud Linear algebra
02.30.Nw Fourier analysis

Quasi-periodic functions on the torus and sl(n)-elliptic Lie algebra

T. Skrypnyk

J. Math. Phys. 53, 023502 (2012); http://dx.doi.org/10.1063/1.3681211 (19 pages)

Online Publication Date: 1 February 2012

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We investigate properties of the sl(n) automorphic elliptic algebra math(sl(n)). We prove it to be math quasi-graded Lie algebra which could be viewed as a deformation of a graded loop algebra. We show that it admits the decomposition into the direct sum of two subalgebras: math(sl(n)) = math(sl(n))++math(sl(n))− consistent with the described quasi-grading. We prove that math(sl(n))±* = math(sl(n))∓, i.e., Lie algebras math(sl(n)), math(sl(n))+, and math(sl(n))− constitute the Manin triple. We explicitly construct a central extension of math(sl(n)). We find its algebra of differentiations and its central extension which coincide with the quasi-graded deformation of the Virasoro algebra.
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02.20.Sv Lie algebras of Lie groups
02.30.Sa Functional analysis

Planar waveguide with “twisted” boundary conditions: Small width

Denis Borisov and Giuseppe Cardone

J. Math. Phys. 53, 023503 (2012); http://dx.doi.org/10.1063/1.3681895 (22 pages)

Online Publication Date: 7 February 2012

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We consider a planar waveguide with “twisted” boundary conditions. By twisting we mean a special combination of Dirichlet and Neumann boundary conditions. Assuming that the width of the waveguide goes to zero, we identify the effective (limiting) operator as the width of the waveguide tends to zero, establishes the uniform resolvent convergence in various possible operator norms, and gives the estimates for the rates of convergence. We show that studying the resolvent convergence can be treated as a certain threshold effect and we present an elegant technique which justifies such point of view.
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02.60.Lj Ordinary and partial differential equations; boundary value problems
84.40.Az Waveguides, transmission lines, striplines
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