Local Perturbations of Quantum Waveguides

We give necessary and sufficient conditions for the occurrence of eigenvalues of the Schrodinger operator in strips and cylinders under small localized perturbations. We construct asymptotic approximations for the eigenvalues. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 3, pp. 358–371, December, 2005.     

Time Delay and Short-Range Scattering in Quantum Waveguides

Although many physical arguments account for using a modified definition of time delay in multichannel-type scattering processes, one can hardly find rigorous results on that issue in the literature. We try to fill in this gap by showing, both in an abstract setting and in a short-range case, the identity of the modified time delay and the Eisenbud-Wigner time delay in waveguides. In the short-range case we also obtain limiting absorption principles, state spectral properties of the total Hamiltonian, prove the existence of the wave operators and show an explicit formula for the S-matrix. The proofs rely on stationary and commutator methods. Communicated by Yosi Avron submitted 12/04/05, accepted 13/05/05     

Absolute Quantum Energy Inequalities in Curved Spacetime

Quantum Energy Inequalities (QEIs) are results which limit the extent to which the smeared renormalized energy density of the quantum field can be negative, when averaged along a timelike curve or over a more general timelike submanifold in spacetime. On globally hyperbolic spacetimes the minimally-coupled massive quantum Klein–Gordon field is known to obey a ‘difference’ QEI that depends on a reference state chosen arbitrarily from the class of Hadamard states. In many spacetimes of interest this bound cannot be evaluated explicitly. In this paper we obtain the first ‘absolute’ QEI for the minimally-coupled massive quantum Klein–Gordon field on four dimensional globally hyperbolic spacetimes; that is, a bound which depends only on the local geometry. The argument is an adaptation of that used to prove the difference QEI and utilizes the Sobolev wave-front set to give a complete characterization of the singularities of the Hadamard series. Moreover, the bound is explicit and can be formulated covariantly under additional (general) conditions. We also generalise our results to incorporate adiabatic states. Dedicated to Klaus Fredenhagen on the occasion of his 60th birthday. Submitted: February 27, 2007. Accepted: November 22, 2007. Calvin J. Smith: Address from 1 January 2007.     

Quantum Scattering in a Periodically Pulsed Magnetic Field

In this paper, we study the quantum dynamics of a charged particle in the plane in the presence of a periodically pulsed magnetic field perpendicular to the plane. We show that by controlling the cycle when the magnetic field is switched on and off appropriately, the result of the asymptotic completeness of wave operators can be obtained under the assumption that the potential V satisfies the decaying condition \({|V(x)| \le C(1 + |x|)^{-\rho}}\) for some \({\rho > 0}\).     

Violation of Symmetry in the System of Three Laterally Coupled Quantum Waveguides,and Resonance Asymptotics

The system of three two-dimensional laterally coupled quantum waveguides with the Dirichlet boundary condition is considered. Violation of geometrical symmetry is studied. The behaviour of the resonance asymptotics and the transition “eigenvalue — resonance” are investigated. Bibliography: 16 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 221–227.     

A Note on the Type of Local von Neumann Algebras in Quantum Field Theory over Curved Space-time

It is shown that there are relations between the Connes invariant for local v. Neumann algebras belonging to different curved space-times provided these local algebras are connected by a certain asymptotic condition. This asymptotic condition is implied by a scaling limit condition as well as by a condition of connectedness by large translations.     

Absolute Continuity in Periodic Waveguides

We study second order elliptic operators with periodic coefficientsin two-dimensional simply connected periodic waveguides withthe Dirichlet or Neumann boundary conditions. It is proved thatunder some mild smoothness restrictions on the coefficients,such operators have purely absolutely continuous spectra. Theproof follows a method suggested previously by A. Morame totackle periodic operators with variable coefficients in dimension2. 2000 Mathematical Subject Classification: 35J10, 35P05, 35J25.     

Asymptotic Solutions of Nonrelativistic Equations of Quantum Mechanics in Curved Nanotubes: I. Reduction to Spatially One-Dimensional Equations

We consider equations of nonrelativistic quantum mechanics in thin three-dimensional tubes (nanotubes). We suggest a version of the adiabatic approximation that permits reducing the original three-dimensional equations to one-dimensional equations for a wide range of energies of longitudinal motion. The suggested reduction method (the operator method for separating the variables) is based on the Maslov operator method. We classify the solutions of the reduced one-dimensional equation. In Part I of this paper, we deal with the reduction problem, consider the main ideas of the operator separation of variables (in the adiabatic approximation), and derive the reduced equations. In Part II, we will discuss various asymptotic solutions and several effects described by these solutions.     

Instanton Effects and Quantum Spectral Curves

We study a spectral problem associated to the quantization of a spectral curve arising in local mirror symmetry. The perturbative WKB quantization condition is determined by the quantum periods, or equivalently by the refined topological string in the Nekrasov–Shatashvili (NS) limit. We show that the information encoded in the quantum periods is radically insufficient to determine the spectrum: there is an infinite series of instanton corrections, which are non-perturbative in \({\hbar}\), and lead to an exact WKB quantization condition. Moreover, we conjecture the precise form of the instanton corrections: they are determined by the standard or unrefined topological string free energy, and we test our conjecture successfully against numerical calculations of the spectrum. This suggests that the non-perturbative sector of the NS refined topological string contains information about the standard topological string. As an application of the WKB quantization condition, we explain some recent observations relating membrane instanton corrections in ABJM theory to the refined topological string.     

Quantum Gravitational Effects on the Boundary

Quantum gravitational effects might hold the key to some of the outstanding problems in theoretical physics. We analyze the perturbative quantum effects on the boundary of a gravitational system and the Dirichlet boundary condition imposed at the classical level. Our analysis reveals that for a black hole solution, there is a contradiction between the quantum effects and the Dirichlet boundary condition: the black hole solution of the one-particle-irreducible action no longer satisfies the Dirichlet boundary condition as would be expected without going into details. The analysis also suggests that the tension between the Dirichlet boundary condition and loop effects is connected with a certain mechanism of information storage on the boundary.     

Nonlinear Wave Equations in Infinite Waveguides

Abstract

We present sharp decay rates as time tends to infinity for solutions to linear Kleinc-Gordon and wave equations in domains with infinite boundaries like infinite waveguides, as well as the global well-posedness and the asymptotics for small data for the solutions to the associated nonlinear initial-boundary value problems.     

Dispersion in Cylindrical Waveguides with Uncertain Parameters

In order to localize cracks in cylindrical structures using guided waves, precise knowledge of the wave speeds is crucial. Instead of basing calculations on crisp parameters, for this Structural Health Monitoring application, uncertainty in parameters is handled by representing parameters as fuzzy numbers and applying the Transformation Method. For calculating dispersion curves, the Waveguide Finite Element Method is used for each parameter set. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Self-focusing Multibump Standing Waves in Expanding Waveguides

Let M be a smooth k-dimensional closed submanifold of \({\mathbb{R}^N, N \geq 2}\), and let Ω _R be the open tubular neighborhood of radius 1 of the expanded manifold \({M_R := \{R_x : x \in M\}}\). For R sufficiently large we show the existence of positive multibump solutions to the problem

$ -\Delta u + \lambda u = f(u)\,{\rm in}\,\Omega_R,\quad u= 0\,{\rm on}\,\partial\Omega_R. $

The function f is superlinear and subcritical, and λ >  ?λ₁, where λ₁ is the first Dirichlet eigenvalue of ?Δ in the unit ball in \({\mathbb{R}^{N-k}}\).