Second-order superintegrable quantum systems

A classical (or quantum) superintegrable system on an n-dimensional Riemannian manifold is an integrable Hamiltonian system with potential that admits 2n ? 1 functionally independent constants of the motion that are polynomial in the momenta, the maximum number possible. If these constants of the motion are all quadratic, then the system is second-order superintegrable, the most tractable case and the one we study here. Such systems have remarkable properties: multi-integrability and separability, a quadratic algebra of symmetries whose representation theory yields spectral information about the Schrödinger operator, and deep connections with expansion formulas relating classes of special functions. For n = 2 and for conformally flat spaces when n = 3, we have worked out the structure of the classical systems and shown that the quadratic algebra always closes at order 6. Here, we describe the quantum analogs of these results. We show that, for nondegenerate potentials, each classical system has a unique quantum extension.     

New Exact Ground States for One-Dimensional Quantum Many-Body Systems

We consider one-dimensional quantum many-body systems with pair interactions in external fields and (re)investigate the conditions under which exact ground-state wave functions of product type can be found. Contrary to a claim in the literature that an exhaustive list of such systems is already known, we show that this list can still be enlarged considerably. In particular, we are able to calculate exact ground-state wave functions for a class of quantum many-body systems with Ax ^–2+Bx ² interaction potentials and external potentials given by sixth-order polynomials.     

Viscous Multipolar Fluids-Physical Background and Mathematical Theory-

We introduce the theory of multipolar fluids in which constitutive laws depend linearly not only on the first spatial gradients of velocity as in classical Navier-Stokes theory of newtonian fluids but also on its higher order spatial gradients up to the order 2k − 1, k = 2, 3,… Such fluids are called k-polar fluids. A thermodynamic theory of the constitutive equations satisfying the second law of thermodynamics and the principle of material frame indifference is developed. Special thermodynamic processes as isothermal, barotropic, adiabatic and general heat-conductive motion for compressible multipolar fluids are studied. It is well known that there does not exist adequate existence theory for compressible newtonian fluids. We given a consistent theory for compressible multipolar fluids in two or three dimensions, i. e. we prove the global in time existence of weak solutions for the initial boundary value problems in bounded domains for the systems of partial differential equations describing isothermal, barotropic, adiabatic and general compressible motion. Under some assumptions on the regularity of the initial data and external forces, we prove existence of strong solutions, uniqueness and regularity. Some other properties as e. g. cavitation of density are discussed. We put stress on the lowest possible polarity of the fluid. In the isothermal case we consider the polarity k ≧ 2 and in barotropic and heat-conductive gas the polarity k ≧ 3.     

Isoperiodic classical systems and their quantum counterparts

One-dimensional isoperiodic classical systems have been first analyzed by Abel. Abel’s characterization can be extended for singular potentials and potentials which are not defined on the whole real line. The standard shear equivalence of isoperiodic potentials can also be extended by using reflection and inversion transformations. We provide a full characterization of isoperiodic rational potentials showing that they are connected by translations, reflections or Joukowski transformations. Upon quantization many of these isoperiodic systems fail to exhibit identical quantum energy spectra. This anomaly occurs at order O(?²) because semiclassical corrections of energy levels of order O(?) are identical for all isoperiodic systems. We analyze families of systems where this quantum anomaly occurs and some special systems where the spectral identity is preserved by quantization. Conversely, we point out the existence of isospectral quantum systems which do not correspond to isoperiodic classical systems.     

Tensor products of quantized tilting modules

LetU _k denote the quantized enveloping algebra corresponding to a finite dimensional simple complex Lie algebraL. Assume that the quantum parameter is a root of unity ink of order at least the Coxeter number forL. Also assume that this order is odd and not divisible by 3 if typeG ₂ occurs. We demonstrate how one can define a reduced tensor product on the familyF consisting of those finite dimensional simpleU _k-modules which are deformations of simpleL and which have non-zero quantum dimension. This together with the work of Reshetikhin-Turaev and Turaev-Wenzl prove that (U _k,F) is a modular Hopf algebra and hence produces invariants of 3-manifolds. Also by recent work of Duurhus, Jakobsen and Nest it leads to a general topological quantum field theory. The method of proof explores quantized analogues of tilting modules for algebraic groups.     

Quantum Tunneling in an Order-Parameter-Preserving Antiferromagnet

We have studied quantum tunneling in an order-parameter-preserving antiferromagnet with the help of Holstein-Primakoff transformation. It is found that, when the system being prepared in a coherent state, there exist the quantum tunneling between lattices k and k+1, k and k−1, respectively. In particular, when the lattice is infinitely long and the spin excitations are in the long-wavelength limit, quantum tunneling disappear between lattices k and k+1, and that k and k−1, in this case the magnetic soliton appears.     

Heisenberg-Integrable Spin Systems

We investigate certain classes of integrable classical or quantum spin systems. The first class is characterized by the recursively defined property P saying that the spin system consists of a single spin or can be decomposed into two uniformly coupled or disjoint subsystems with property P. For these systems the time evolution can be explicitly calculated. The second class consists of spin systems where all non-zero coupling constants have the same strength (spin graphs) possessing N − 1 independent, commuting constants of motion of Heisenberg type. These systems are shown to have the above property P and can be characterized as spin graphs not containing chains of length four as vertex-induced sub-graphs. We completely enumerate and characterize all spin graphs up to N = 5 spins. Applications to the construction of symplectic numerical integrators for non-integrable spin systems are briefly discussed.      

Canonical transformations and exact invariants for time‐dependent Hamiltonian systems

An exact invariant is derived for n‐degree‐of‐freedom non‐relativistic Hamiltonian systems with general time‐dependent potentials. To work out the invariant, an infinitesimalcanonical transformation is performed in the framework of the extended phase‐space. We apply this approach to derive the invariant for a specific class of Hamiltonian systems. For the considered class of Hamiltonian systems, the invariant is obtained equivalently performing in the extended phase‐space a finitecanonical transformation of the initially time‐dependent Hamiltonian to a time‐independent one. It is furthermore shown that the invariant can be expressed as an integral of an energy balance equation. The invariant itself contains a time‐dependent auxiliary function ξ (t) that represents a solution of a linear third‐order differential equation, referred to as the auxiliary equation. The coefficients of the auxiliary equation depend in general on the explicitly known configuration space trajectory defined by the system's time evolution. This complexity of the auxiliary equation reflects the generally involved phase‐space symmetry associated with the conserved quantity of a time‐dependent non‐linear Hamiltonian system. Our results are applied to three examples of time‐dependent damped and undamped oscillators. The known invariants for time‐dependent and time‐independent harmonic oscillators are shown to follow directly from our generalized formulation.     

Coupled N-dimensional harmonic oscillator systems

A general decoupling procedure is described for systems of N interacting harmonic oscillators coupled by some bilinear perturbing potentials, and explicit conditions are derived which must be fulfilled in order to obtain physical bound states. Illustrative examples include a general treatment of the 2-oscillator system, some particular extensions to systems containing 3 oscillators, and a class of N-oscillator models.     

Transport across an Anderson quantum dot in the intermediate coupling regime

We describe linear and nonlinear transport across a strongly interacting single impurity Anderson model quantum dot with intermediate coupling to the leads, i.e. with tunnel coupling Γ of the order of the thermal energy k _B T. The coupling is large enough that sequential tunneling processes (second order in the tunneling Hamiltonian) alone do not suffice to properly describe the transport characteristics. Upon applying a density matrix approach, the current is expressed in terms of rates obtained by considering a very small class of diagrams which dress the sequential tunneling processes by charge fluctuations. We call this the “dressed second order” (DSO) approximation. One advantage of the DSO is that, still in the Coulomb blockade regime, it can describe the crossover from thermally broadened to tunneling broadened conductance peaks. When the temperature is decreased even further (k _B T < Γ), the DSO captures Kondesque behaviours of the Anderson quantum dot qualitatively: we find a zero bias anomaly of the differential conductance versus applied bias, an enhancement of the conductance with decreasing temperature as well as universality of the shape of the conductance as function of the temperature. We can without complications address the case of a spin degenerate level split energetically by a magnetic field. In case spin dependent chemical potentials are assumed and only one of the four chemical potentials is varied, the DSO yields in principle only one resonance. This seems to be in agreement with experiments with pseudo spin [U. Wilhelm, J. Schmid, J. Weis, K.V. Klitzing, Physica E 14, 385 (2002)]. Furthermore, we get qualitative agreement with experimental data showing a cross-over from the Kondo to the empty orbital regime.     

Quantum completely integrable systems connected with semi-simple Lie algebras

The quantum version of the dynamical systems whose integrability is related to the root systems of semi-simple Lie algebras are considered. It is proved that the operators _k introduced by Calogero et al. are integrals of motion and that they commute. The explicit form of another class of integrals of motion is given for systems with few degrees of freedom.     

Quasiclassical symmetry properties II

Proofs are given that the quasiclassical approach proposed previously is able to work in conjunction with the 1/N method. Accordingly, the expansion parameter k = 2l + N − a of the shifted 1/N method should be chosen, order by order, such that the sum of corrections to the zeroth-order result vanishes. This interconnection criterion leads to order-dependent algebraic equations for the parameter k. In turn, the underlying phase-space quantum becomes just half the parameter k implied in this way. Alternative fixing conditions, based on the selection of a dominant potential term, can also be proposed. Quasiclassical symmetry transformations leaving corresponding equivalence classes of Hamiltonian forms invariant are established. Then energy levels and couplings characterizing such Hamiltonians become subject to mutual conversions. Scaling properties of the phase-space quantum are discussed. In addition, this quantum exhibits a covariance behaviour with respect to the quasiclassical symmetry transformations mentioned above. Critical coupling for several short-range potentials are given to first order. Generalizations to resonances and nonlinear quantum-mechanical potentials are also made. Except for dyons, we restrict ourselves to spherically symmetric nonrelativistic Hamiltonians.     

量子Generalized Reed-Solomon码

构造出了一族量子纠错码,这族码具有参数［［n,n-2k,k+1］］_q,是q维量子系统上的码,q是任意素数的幂.这族码的最小距离达到了理论上限,因此,以码距来说,它是最优的.证明了当2≤n≤q或者q²-q+2≤n≤q²时,码都是存在的. 关键词： 量子Generalized Reed-Solomon码 量子MDS码 量子纠错码 量子信息     

Galilean-Invariant (2+1)-Dimensional Models with a Chern–Simons-Like Term andD=2 Noncommutative Geometry

We consider a newD=2 nonrelativistic classical mechanics model providing via the Noether theorem the (2+1)-Galilean symmetry algebra with two central charges: massmand the coupling constantkof a Chern–Simons-like term. In this way we provide the dynamical interpretation of the second central charge of the (2+1)-dimensional Galilean algebra. We discuss also the interpretation ofkas describing the noncommutativity ofD=2 space coordinates. The model is quantized in two ways: using the Ostrogradski–Dirac formalism for higher order Lagrangians with constraints and the Faddeev–Jackiw method which describes constrained systems and produces nonstandard symplectic structures. We show that our model describes the superposition of a free motion in noncommutativeD=2 spaces as well as the “internal” oscillator modes. We add a suitably chosen class of velocity-dependent two-particle interactions, which is described by local potentials inD=2 noncommutative space. We treat, in detail, the particular case of a harmonic oscillator and describe its quantization. It appears that the indefinite metric due to the third order time derivative term in the field equations, even in the presence of interactions, can be eliminated by the imposition of a subsidiary condition.     

Luminescence decays with underlying distributions: General properties and analysis with mathematical functions

In this work, an analysis of the general properties of the luminescence decay law is carried out. The conditions that a luminescence decay law must satisfy in order to correspond to a probability density function of rate constants are established. From an analysis of the general form of the luminescence decay law, it is concluded that the decay must be either exponential or sub-exponential for all times, in order to be represented by a distribution of rate constants H(k). Sub-exponentiality is nevertheless not a sufficient condition. Only decays that are completely monotonic have a probability density function of rate constants. The construction of the decay function from cumulant and moment expansions is studied, as well as the corresponding calculation of H(k) from a cumulant expansion. The asymptotic behavior of the decay laws is considered in detail, and the relation between this behavior and the form of H(k) for small k is explored. Several generalizations of the exponential decay function, namely the Kohlrausch, Becquerel, Mittag-Leffler and Heaviside decay functions, as well as the Weibull and truncated Gaussian rate constant distributions are discussed.     

Hidden symmetry breaking and the Haldane phase inS=1 quantum spin chains

We study the phase diagram ofS=1 antiferromagnetic chains with particular emphasis on the Haldane phase. The hidden symmetry breaking measured by the string order parameter of den Nijs and Rommelse can be transformed into an explicit breaking of aZ ₂×Z ₂ symmetry by a nonlocal unitary transformation of the chain. For a particular class of Hamiltonians which includes the usual Heisenberg Hamiltonian, we prove that the usual Néel order parameter is always less than or equal to the string order parameter. We give a general treatment of rigorous perturbation theory for the ground state of quantum spin systems which are small perturbations of diagonal Hamiltonians. We then extend this rigorous perturbation theory to a class of diagonally dominant Hamiltonians. Using this theory we prove the existence of the Haldane phase in an open subset of the parameter space of a particular class of Hamiltonians by showing that the string order parameter does not vanish and the hiddenZ ₂×Z ₂ symmetry is completely broken. While this open subset does not include the usual Heisenberg Hamiltonian, it does include models other than VBS models.     

Deformation potentials and photo-response of strained PbSe quantum wells and quantum dots

Deformation potentials (D_u and D_d) for PbSe where analyzed using transmission data of PbSe/PbEuSeTe multi-quantum wells (MQWs). We use calculations based on a k·p model to obtain the strain induced intervalley splitting in the quantum wells. For the reduction of the Fabri–Pérot interference fringes of the multilayer structures we design a PbEuSeTe/BaF₂ anti-reflex coating which is deposited on top of the MQWs. At low temperature we found PbSe deformation potentials D_u=-2.36 and 5.88. The results of the transmission measurements are compared with photo-current spectra measured with self-assembled PbSe/PbEuTe quantum dot superlattices.     

Natural Star Products on Symplectic Manifolds and Quantum Moment Maps

We define a natural class of star products: those which are given by a series of bidifferential operators which at order k in the deformation parameter have at most k derivatives in each argument. This class includes all the standard constructions of star products. We show that any such star product on a symplectic manifold defines a unique symplectic connection. We parametrise such star products, study their invariance properties and give necessary and sufficient conditions for them to have a quantum moment map. We show that Kravchenko's sufficient condition for a moment map for a Fedosov star product is also necessary.     

Path integral approach for superintegrable potentials on spaces of nonconstant curvature: I. Darboux spaces <Emphasis Type="Italic">D</Emphasis><Subscript>I</Subscript> and <Emphasis Type="Italic">D</Emphasis><Subscript>II</Subscript>

In this paper, the Feynman path integral technique is applied for superintegrable potentials on two-dimensional spaces of nonconstant curvature: these spaces are Darboux spaces D _I and D _II. On D _I, there are three, and on D _II four such potentials. We are able to evaluate the path integral in most of the separating coordinate systems, leading to expressions for the Green functions, the discrete and continuous wave-functions, and the discrete energy-spectra. In some cases, however, the discrete spectrum cannot be stated explicitly, because it is either determined by a transcendental equation involving parabolic cylinder functions (Darboux space I), or by a higher order polynomial equation. The solutions on D _I in particular show that superintegrable systems are not necessarily degenerate. We can also show how the limiting cases of flat space (constant curvature zero) and the two-dimensional hyperboloid (constant negative curvature) emerge. The text was submitted by the authors in English.