Volume 46, Issue 5, May 2005
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Exact solutions of two complementary onedimensional quantum manybody systems on the halfline
View Description Hide DescriptionWe consider two particular onedimensional quantum manybody systems with local interactions related to the root system. Both models describe identical particles moving on the halfline with nontrivial boundary conditions at the origin, but in the first model the particles interact with the delta interaction while in the second via a particular momentum dependent interaction commonly known as deltaprime interaction. We show that the Bethe ansatz solution of the deltainteraction model is consistent even for the general case where the particles are distinguishable, whereas for the deltaprime interaction it only is consistent and nontrivial in the fermion case. We also establish a duality between the bosonic delta and the fermionic deltaprime model, and we elaborate on the physical interpretations of these models.

Analytic controllability of timedependent quantum control systems
View Description Hide DescriptionThe question of controllability is investigated for a quantum control system in which the Hamiltonian operator components carry explicit time dependence which is not under the control of an external agent. We consider the general situation in which the state moves in an infinitedimensional Hilbert space, a drift term is present, and the operators driving the state evolution may be unbounded. However, considerations are restricted by the assumption that there exists an analytic domain, dense in the state space, on which solutions of the controlled Schrödinger equation may be expressed globally in exponential form. The issue of controllability then naturally focuses on the ability to steer the quantum state on a finitedimensional submanifold of the unit sphere in Hilbert space—and thus on analytic controllability. A relatively straightforward strategy allows the extension of Liealgebraic conditions for strong analytic controllability derived earlier for the simpler, timeindependent system in which the drift Hamiltonian and the interaction Hamiltonian have no intrinsic time dependence. Enlarging the state space by one dimension corresponding to the time variable, we construct an augmented control system that can be treated as time independent. Methods developed by Kunita can then be implemented to establish controllability conditions for the onedimensionreduced system defined by the original timedependent Schrödinger control problem. The applicability of the resulting theorem is illustrated with selected examples.

Control aspects of holonomic quantum computation
View Description Hide DescriptionA unifying framework for the control of quantum systems with nonAbelian holonomy is presented. It is shown that, from a control theoretic point of view, holonomic quantum computation can be treated as a control system evolving on a principal fiber bundle. An extension of methods developed for these classical systems may be applied to quantum holonomic systems to obtain insight into the control properties of such systems and to construct control algorithms for two established examples of the computing paradigm.

Stability of three unit charges: Necessary conditions
View Description Hide DescriptionWe consider the stability of three Coulomb charges with finite masses in the framework of nonrelativistic quantum mechanics. A simple physical condition on masses is derived to guarantee the absence of bound states below the dissociation thresholds. In particular this proves that certain negative muonic ions are unstable, thus extending the old result of Thirring to the actual values of all masses. The proof is done by reducing the initial problem to the question of binding of one particle in some effective potential.

Strict deformation quantization for a particle in a magnetic field
View Description Hide DescriptionRecently, we introduced a mathematical framework for the quantization of a particle in a variable magnetic field. It consists in a modified form of the Weyl pseudodifferential calculus and a algebraic setting, these two points of view being isomorphic in a suitable sense. In the present paper we leave Planck’s constant vary, showing that one gets a strict deformation quantization in the sense of Rieffel. In the limit one recovers a Poisson algebra induced by a symplectic form defined in terms of the magnetic field.

Algebraic methods for diagonalization of a quaternion matrix in quaternionic quantum theory
View Description Hide DescriptionBy means of complex representation and real representation of a quaternion matrix, this paper studies the problem of diagonalization of a quaternion matrix, gives two algebraic methods for diagonalization of quaternion matrices in quaternionic quantum theory.

Symmetric informationally complete–positive operator valued measures and the extended Clifford group
View Description Hide DescriptionWe describe the structure of the extended Clifford group [defined to be the group consisting of all operators, unitary and antiunitary, which normalize the generalized Pauli group (or Weyl–Heisenberg group as it is often called)]. We also obtain a number of results concerning the structure of the Clifford group proper (i.e., the group consisting just of the unitary operators which normalize the generalized Pauli group). We then investigate the action of the extended Clifford group operators on symmetric informationally complete–positive operator valued measures (or SIC–POVMs) covariant relative to the action of the generalized Pauli group. We show that each of the fiducial vectors which has been constructed so far (including all the vectors constructed numerically by Renes et al.) is an eigenvector of one of a special class of order 3 Clifford unitaries. This suggests a strengthening of a conjecture of Zauner’s. We give a complete characterization of the orbits and stability groups in dimensions 2–7. Finally, we show that the problem of constructing fiducial vectors may be expected to simplify in the infinite sequence of dimensions 7,13,19,21,31,… . We illustrate this point by constructing exact expressions for fiducial vectors in dimensions 7 and 19.

Mathematical analysis of the Mandelstam–Tamm timeenergy uncertainty principle
View Description Hide DescriptionIn the Mandelstam–Tamm version of the timeenergy uncertainty principle denotes the infimum of time intervals that elapse before the change in the mean of any observable has the same magnitude as its standard deviation. We clarify this interpretation, and show that the infimum is achieved for certain observables and thus that this famous inequality is actually an equality.

Saturated Kochen–Speckertype configuration of 120 projective lines in eightdimensional space and its group of symmetry
View Description Hide DescriptionThere exists an example of a set of 40 projective lines in eightdimensional Hilbert space producing a Kochen–Speckertype contradiction. This set corresponds to a known nohidden variables argument due to Mermin. In the present paper it is proved that this set admits a finite saturation, i.e., an extension up to a finite set with the following property: every subset of pairwise orthogonal projective lines has a completion, i.e., is contained in at least one subset of eight pairwise orthogonal projective lines. An explicit description of such an extension consisting of 120 projective lines is given. The idea to saturate the set of projective lines related to Mermin’s example together with the possibility to have a finite saturation allow to find the corresponding group of symmetry. This group is described explicitely and is shown to be generated by reflections. The natural action of the mentioned group on the set of all subsets of pairwise orthogonal projective lines of the mentioned extension is investigated. In particular, the restriction of this action to complete subsets is shown to have only four orbits, which have a natural characterization in terms of the construction of the saturation.

Some useful combinatorial formulas for bosonic operators
View Description Hide DescriptionWe give a general expression for the normally ordered form of a function where is a function of boson creation and annihilation operators satisfying . The expectation value of this expression in a coherent state becomes an exact generating function of Feynmantype graphs associated with the zerodimensional quantum field theory defined by . This enables one to enumerate explicitly the graphs of given order in the realm of combinatorially defined sequences. We give several examples of the use of this technique, including the applications to Kerrtype and superfluiditytype Hamiltonians.

Vortices set and the applied magnetic field for superconductivity in dimension three
View Description Hide DescriptionIn this paper, the structure of vortices set of the Ginzburg–Landau system of the superconductivity in dimension three was studied when applied magnetic field. This singularities set is onedimensional rectifiable. Its generalized mean curvature was given.

Strongelectricfield eigenvalue asymptotics for the Iwatsuka model
View Description Hide DescriptionWe consider the twodimensional Schrödinger operator, , where is a nonnegative scalar potential decaying at infinity like , and is a magnetic vector potential. Here, is of the form and the magnetic field is assumed to be positive, bounded, and monotonically increasing on (the Iwatsuka model). Following the argument as in Refs. 15, 16, and 17 [Raikov, G. D., Lett. Math. Phys., 21, 41–49 (1991); Raikov, G. D, Commun. Math. Phys., 155, 415–428 (1993); Raikov, G. D. Asymptotic Anal., 16, 87–89 (1998)], we obtain the asymptotics of the number of discrete spectra of crossing a real number in the gap of the essential spectrum as the coupling constant tends to , respectively.
