Volume 55, Issue 12, December 2014
Index of content:

We present a new complete set of states for a class of open quantum systems, to be used in expansion of the Green’s function and the timeevolution operator. A remarkable feature of the complete set is that it observes timereversal symmetry in the sense that it contains decaying states (resonant states) and growing states (antiresonant states) parallelly. We can thereby pinpoint the occurrence of the breaking of timereversal symmetry at the choice of whether we solve Schrödinger equation as an initialcondition problem or a terminalcondition problem. Another feature of the complete set is that in the subspace of the central scattering area of the system, it consists of contributions of all states with point spectra but does not contain any background integrals. In computing the time evolution, we can clearly see contribution of which point spectrum produces which time dependence. In the whole infinite state space, the complete set does contain an integral but it is over unperturbed eigenstates of the environmental area of the system and hence can be calculated analytically. We demonstrate the usefulness of the complete set by computing explicitly the survival probability and the escaping probability as well as the dynamics of wave packets. The origin of each term of matrix elements is clear in our formulation, particularly, the exponential decays due to the resonance poles.
 ARTICLES

 Partial Differential Equations

The existence of positive solutions with prescribed L ^{2}norm for nonlinear Choquard equations
View Description Hide DescriptionIn this paper, we study the existence of positive solutions with prescribed L ^{2}norm to a class of nonlinear Choquard equation −Δu − λu = (I α ∗F(u)) F′(u) in ℝ^{ N }, where λ ∈ ℝ, N ≥ 3, α ∈ (0, N), I α :ℝ^{ N } → ℝ is the Riesz potential. Under some conditions imposed on F, by using a minimax procedure and the concentration compactness of P. L. Lions, we show that for any c > 0, the equation possesses at least a couple (uc , λc ) ∈ H ^{1}(ℝ^{ N }) × ℝ^{−} of weak solution such that .

Opening up and control of spectral gaps of the Laplacian in periodic domains
View Description Hide DescriptionThe main result of this work is as follows: for arbitrary pairwise disjoint, finite intervals (αj , βj ) ⊂ [0, ∞), j = 1, …, m, and for arbitrary n ≥ 2, we construct a family of periodic noncompact domains {Ω^{ε}⊂ℝ^{ n }}ε>0 such that the spectrum of the Neumann Laplacian in Ω^{ε} has at least m gaps when ε is small enough, moreover the first m gaps tend to the intervals (αj , βj ) as ε → 0. The constructed domain Ω^{ε} is obtained by removing from ℝ^{ n } a system of periodically distributed “traplike” surfaces. The parameter ε characterizes the period of the domain Ω^{ε}, also it is involved in a geometry of the removed surfaces.

Bounded Sobolev norms for KleinGordon equations under nonresonant perturbation
View Description Hide DescriptionIn this paper, we prove Anderson localization for the KleinGordon operator on the circle 𝕋 under nonresonant perturbations. Furthermore, using the result, we show that the Sobolev norms of solutions to the corresponding KleinGordon equations remain bounded for all time.

Decay rate estimates for a class of quasilinear hyperbolic equations with damping terms involving pLaplacian
View Description Hide DescriptionIn this paper, we are concerned with the asymptotic behaviour of weak solutions to the initial boundary value problem for a class of quasilinear hyperbolic equations with damping terms involving pLaplacian. By using the multiplier methods, we investigate the stability of weak solutions to the initial boundary value problem and obtain explicit decay rate estimation depending on straincaused stress term and damping terms.

On integration of a multidimensional version of nwave type equation
View Description Hide DescriptionWe represent a version of multidimensional quasilinear partial differential equation (PDE) together with large manifold of particular solutions given in an integral form. The dimensionality of constructed PDE can be arbitrary. We call it the nwave type PDE, although the structure of its nonlinearity differs from that of the classical completely integrable (2+1)dimensional nwave equation. The richness of solution space to such a PDE is characterized by a set of arbitrary functions of several variables. However, this richness is not enough to provide the complete integrability, which is shown explicitly. We describe a class of multisolitary wave solutions in details. Among examples of explicit particular solutions, we represent a lumplattice solution depending on five independent variables. In Appendix, as an important supplemental material, we show that our nonlinear PDE is reducible from the more general multidimensional PDE which can be derived using the dressing method based on the linear integral equation with the kernel of a special type (a modification of the problem). The dressing algorithm gives us a key for construction of higher order PDEs, although they are not discussed in this paper.

A note on the regularity of the solutions to the NavierStokes equations via the gradient of one velocity component
View Description Hide DescriptionWe present a regularity criterion for the solutions to the NavierStokes equations based on the gradient of one velocity component. Starting with the method developed by Cao and Titi [“Global regularity criterion for the 3D NavierStokes equations involving one entry of the velocity gradient tensor,” Arch. Ration. Mech. Anal. 202, 919–932 (2011)] for the case of one entry of the velocity gradient and using further some inequalities concerning the anisotropic Sobolev spaces, we show as a main result that a weak solution u is regular on (0, T), T > 0, provided that ∇u 3 ∈ L^{t} (0, T; L^{s} ), where 2/t + 3/s = 3/2 + 3/(4s) and s ∈ (3/2, 2). It improves the known results for s ∈ (3/2, 15/8).
 Representation Theory and Algebraic Methods

Nontrivial central extensions of 3Lie algebras
View Description Hide DescriptionWe construct a series of infinitedimensional 3Lie algebras which include w ∞ and SDiff(T ^{2}) 3Lie algebras as subalgebras, and determine their nontrivial onedimensional central extensions.

Quantization of borderline Levi conjugacy classes of orthogonal groups
View Description Hide DescriptionWe construct an equivariant quantization of a special family of Levi conjugacy classes of the complex orthogonal group SO(N), whose stabilizer contains a Cartesian factor SO(2) × SO(P), 1 ⩽ P < N, P ≡ N mod 2.
 ManyBody and Condensed Matter Physics

Quantization of interface currents
View Description Hide DescriptionAt the interface of two twodimensional quantum systems, there may exist interface currents similar to edge currents in quantum Hall systems. It is proved that these interface currents are macroscopically quantized by an integer that is given by the difference of the Chern numbers of the two systems. It is also argued that at the interface between two timereversal invariant systems with halfinteger spin, one of which is trivial and the other nontrivial, there are dissipationless spinpolarized interface currents.
 Quantum Mechanics

Curvatureinduced bound states in Robin waveguides and their asymptotical properties
View Description Hide DescriptionWe analyze bound states of Robin Laplacian in infinite planar domains with a smooth boundary, in particular, their relations to the geometry of the latter. The domains considered have locally straight boundary being, for instance, locally deformed halfplanes or wedges, or infinite strips, alternatively they are the exterior of a bounded obstacle. In the situation when the Robin condition is strongly attractive, we derive a twoterm asymptotic formula in which the nexttoleading term is determined by the extremum of the boundary curvature. We also discuss the nonasymptotic case of attractive boundary interaction and show that the discrete spectrum is nonempty if the domain is a local deformation of a halfplane or a wedge of angle less than π, and it is void if the domain is concave.

Gaussian distributions, Jacobi group, and SiegelJacobi space
View Description Hide DescriptionLet be the space of Gaussian distribution functions over ℝ, regarded as a 2dimensional statistical manifold parameterized by the mean μ and the deviation σ. In this paper, we show that the tangent bundle of , endowed with its natural Kähler structure, is the SiegelJacobi space appearing in the context of Number Theory and Jacobi forms. Geometrical aspects of the SiegelJacobi space are discussed in detail (completeness, curvature, group of holomorphic isometries, space of Kähler functions, and relationship to the Jacobi group), and are related to the quantum formalism in its geometrical form, i.e., based on the Kähler structure of the complex projective space. This paper is a continuation of our previous work [M. Molitor, “Remarks on the statistical origin of the geometrical formulation of quantum mechanics,” Int. J. Geom. Methods Mod. Phys. 9(3), 1220001, 9 (2012); M. Molitor, “Information geometry and the hydrodynamical formulation of quantum mechanics,” eprint arXiv (2012); M. Molitor, “Exponential families, Kähler geometry and quantum mechanics,” J. Geom. Phys. 70, 54–80 (2013)], where we studied the quantum formalism from a geometric and informationtheoretical point of view.

Construction of the BarutGirardello type of coherent states for PöschlTeller potential
View Description Hide DescriptionThe PöschlTeller (PT) potential occupies a privileged place among the anharmonic oscillator potentials due to its applications from quantum mechanics to diatomic molecules. For this potential, a polynomial su(1, 1) algebra has been constructed previously. So far, the coherent states (CSs) associated with this algebra have never appeared. In this paper, we construct the coherent states of the BarutGirardello coherent states (BGCSs) type for the PT potentials, which have received less attention in the scientific literature. We obtain these CSs and demonstrate that they fulfil all conditions required by the coherent state. The Mandel parameter for the pure BGCSs and Husimi’s and Pquasi distributions (for the mixedthermal states) is also presented. Finally, the exponential form of the BGCSs for the PT potential has been presented and enabled us to build Perelomov type CSs for the PT potential. We point out that the BGCSs and the Perelomov type coherent states (PCSs) are related via Laplace transform.

Existence of the StarkWannier quantum resonances
View Description Hide DescriptionIn this paper, we prove the existence of the StarkWannier quantum resonances for onedimensional Schrödinger operators with smooth periodic potential and small external homogeneous electric field. Such a result extends the existence result previously obtained in the case of periodic potentials with a finite number of open gaps.

SU(4) based classification of fourlevel systems and their semiclassical solution
View Description Hide DescriptionWe present a systematic method to classify the fourlevel system using SU (4) symmetry as the basis group. It is shown that this symmetry allows three dipole transitions which eventually leads to six possible configurations of the fourlevel system. Using a dressed atom approach, the semiclassical version of each configuration is exactly solved under rotating wave approximation and the symmetry among the Rabi oscillation among various models is studied.

Timereversal symmetric resolution of unity without background integrals in open quantum systems
View Description Hide DescriptionWe present a new complete set of states for a class of open quantum systems, to be used in expansion of the Green’s function and the timeevolution operator. A remarkable feature of the complete set is that it observes timereversal symmetry in the sense that it contains decaying states (resonant states) and growing states (antiresonant states) parallelly. We can thereby pinpoint the occurrence of the breaking of timereversal symmetry at the choice of whether we solve Schrödinger equation as an initialcondition problem or a terminalcondition problem. Another feature of the complete set is that in the subspace of the central scattering area of the system, it consists of contributions of all states with point spectra but does not contain any background integrals. In computing the time evolution, we can clearly see contribution of which point spectrum produces which time dependence. In the whole infinite state space, the complete set does contain an integral but it is over unperturbed eigenstates of the environmental area of the system and hence can be calculated analytically. We demonstrate the usefulness of the complete set by computing explicitly the survival probability and the escaping probability as well as the dynamics of wave packets. The origin of each term of matrix elements is clear in our formulation, particularly, the exponential decays due to the resonance poles.
 Quantum Information and Computation

Positivity, discontinuity, finite resources, and nonzero error for arbitrarily varying quantum channels
View Description Hide DescriptionThis work is motivated by a quite general question: Under which circumstances are the capacities of information transmission systems continuous? The research is explicitly carried out on finite arbitrarily varying quantum channels (AVQCs). We give an explicit example that answers the recent question whether the transmission of messages over AVQCs can benefit from assistance by distribution of randomness between the legitimate sender and receiver in the affirmative. The specific class of channels introduced in that example is then extended to show that the unassisted capacity does have discontinuity points, while it is known that the randomnessassisted capacity is always continuous in the channel. We characterize the discontinuity points and prove that the unassisted capacity is always continuous around its positivity points. After having established shared randomness as an important resource, we quantify the interplay between the distribution of finite amounts of randomness between the legitimate sender and receiver, the (nonzero) probability of a decoding error with respect to the average error criterion and the number of messages that can be sent over a finite number of channel uses. We relate our results to the entanglement transmission capacities of finite AVQCs, where the role of shared randomness is not yet well understood, and give a new sufficient criterion for the entanglement transmission capacity with randomness assistance to vanish.

How to efficiently select an arbitrary Clifford group element
View Description Hide DescriptionWe give an algorithm which produces a unique element of the Clifford group on n qubits ( ) from an integer (the number of elements in the group). The algorithm involves O(n ^{3}) operations and provides, in addition to a canonical mapping from the integers to group elements g, a factorization of g into a sequence of at most 4n symplectic transvections. The algorithm can be used to efficiently select random elements of which are often useful in quantum information theory and quantum computation. We also give an algorithm for the inverse map, indexing a group element in time O(n ^{3}).

Entanglement and the threedimensionality of the Bloch ball
View Description Hide DescriptionWe consider a very natural generalization of quantum theory by letting the dimension of the Bloch ball be not necessarily three. We analyze bipartite state spaces where each of the components has a ddimensional Euclidean ball as state space. In addition to this, we impose two very natural assumptions: the continuity and reversibility of dynamics and the possibility of characterizing bipartite states by local measurements. We classify all these bipartite state spaces and prove that, except for the quantum twoqubit state space, none of them contains entangled states. Equivalently, in any of these nonquantum theories, interacting dynamics is impossible. This result reveals that “existence of entanglement” is the requirement with minimal logical content which singles out quantum theory from our family of theories.

History dependent quantum random walks as quantum lattice gas automata
View Description Hide DescriptionQuantum Random Walks (QRW) were first defined as oneparticle sectors of Quantum Lattice Gas Automata (QLGA). Recently, they have been generalized to include history dependence, either on previous coin (internal, i.e., spin or velocity) states or on previous position states. These models have the goal of studying the transition to classicality, or more generally, changes in the performance of quantum walks in algorithmic applications. We show that several history dependent QRW can be identified as oneparticle sectors of QLGA. This provides a unifying conceptual framework for these models in which the extra degrees of freedom required to store the history information arise naturally as geometrical degrees of freedom on the lattice.

Positionmomentum uncertainty relations in the presence of quantum memory
View Description Hide DescriptionA prominent formulation of the uncertainty principle identifies the fundamental quantum feature that no particle may be prepared with certain outcomes for both position and momentum measurements. Often the statistical uncertainties are thereby measured in terms of entropies providing a clear operational interpretation in information theory and cryptography. Recently, entropic uncertainty relations have been used to show that the uncertainty can be reduced in the presence of entanglement and to prove security of quantum cryptographic tasks. However, much of this recent progress has been focused on observables with only a finite number of outcomes not including Heisenberg’s original setting of position and momentum observables. Here, we show entropic uncertainty relations for general observables with discrete but infinite or continuous spectrum that take into account the power of an entangled observer. As an illustration, we evaluate the uncertainty relations for position and momentum measurements, which is operationally significant in that it implies security of a quantum key distribution scheme based on homodyne detection of squeezed Gaussian states.