Table of contents

It is pointed out that an exactly solvable permutation operator, viewed as the quantization of cyclic shifts, is useful in constructing a basis in which to study the quantum baker's map, a paradigm system of quantum chaos. In the basis of this operator the eigenfunctions of the quantum baker's map are compressed by factors of around five or more. We show explicitly its connection to an operator that is closely related to the usual quantum baker's map. This permutation operator has interesting connections to the art of shuffling cards as well as to the quantum factoring algorithm of Shor via the quantum order finding one. Hence we point out that this well-known quantum algorithm makes crucial use of a quantum chaotic operator, or at least one that is close to the quantization of the left-shift, a closeness that we also explore quantitatively.

We review several results related to the problem of a quantum particle in a random environment. In an introductory part, we recall how several functionals of Brownian motion arise in the study of electronic transport in weakly disordered metals (weak localization). Two aspects of the physics of the one-dimensional strong localization are reviewed: some properties of the scattering by a random potential (time delay distribution) and a study of the spectrum of a random potential on a bounded domain (the extreme value statistics of the eigenvalues). Then we mention several results concerning the diffusion on graphs, and more generally the spectral properties of the Schrödinger operator on graphs. The interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated. Finally, we consider a two-dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices. We recall the connection between spectral properties of this model and winding functionals of planar Brownian motion.

The damping of vortex cyclotron modes is investigated within a generalized quantum theory of vortex waves. Similarly to the case of Kelvin modes, the friction coefficient turns out to be essentially unchanged under such oscillations, but it is shown to be affected by appreciable memory corrections. On the other hand, the non-equilibrium dynamics of the vortex energy, which is investigated within the framework of linear response theory, shows that its memory corrections are negligible. The vortex response is found to be of the Debye type, with a relaxation frequency whose dependence on temperature and impurity concentration reflects the complexity of the heat bath and its interaction with the vortex.

The most general conformally invariant bending energy of a closed four-dimensional surface, polynomial in the extrinsic curvature and its derivatives, is constructed. This invariance manifests itself as a set of constraints on the corresponding stress tensor. If the topology is fixed, there are three independent polynomial invariants: two of these are the straightforward quartic analogues of the quadratic Willmore energy for a two-dimensional surface; one is intrinsic (the Weyl invariant), the other extrinsic; the third invariant involves a sum of a quadratic in gradients of the extrinsic curvature—which is not itself invariant—and a quartic in the curvature. The four-dimensional energy quadratic in extrinsic curvature plays a central role in this construction.

We present a family of birational transformations in CP₂ depending on two, or three, parameters which do not, generically, preserve meromorphic 2-forms. With the introduction of the orbit of the critical set (vanishing condition of the Jacobian), also called the 'post-critical set', we get some new structures, some 'non-analytic' 2-form which reduce to meromorphic 2-forms for particular subvarieties in the parameter space. On these subvarieties, the iterates of the critical set have a polynomial growth in the degrees of the parameters, while one has an exponential growth out of these subspaces. The analysis of our birational transformation in CP₂ is first carried out using the Diller–Favre criterion in order to find the complexity reduction of the mapping. The integrable cases are found. The identification between the complexity growth and the topological entropy is, once again, verified. We perform plots of the post-critical set, as well as calculations of Lyapunov exponents for many orbits, confirming that generically no meromorphic 2-form can be preserved for this mapping. These birational transformations in CP₂, which, generically, do not preserve any meromorphic 2-form, are extremely similar to other birational transformations we previously studied, which do preserve meromorphic 2-forms. We note that these two sets of birational transformations exhibit totally similar results as far as topological complexity is concerned, but drastically different results as far as a more 'probabilistic' approach of dynamical systems is concerned (Lyapunov exponents). With these examples we see that the existence of a preserved meromorphic 2-form explains most of the (numerical) discrepancies between the topological and probabilistic approaches of dynamical systems.

Using the Wigner transform, we present an alternative derivation of the partial differential equation satisfied by the Slater sum, which is the diagonal element of the canonical Bloch density matrix. This is done in one dimension for the case of a general confining potential and also for the case of N independent fermions harmonically confined in d dimensions. We also present a simple proof of the so-called differential virial theorem for each of these cases.

Problems posed by semirelativistic Hamiltonians of the form are studied. It is shown that energy upper bounds can be constructed in terms of certain related Schrödinger operators; these bounds include free parameters which can be chosen optimally.

Integral transforms which map functions on onto their integrals on circular cones having fixed axis direction and variable opening angle are introduced and studied as generalizations of the known Radon transform. Besides their intrinsic mathematical interest, they serve as backbone support to emission imaging based on Compton scattered radiation, the way the standard Radon transform does for emission imaging based on non-scattered radiation. In this work, we establish its basic properties and prove analytically its invertibility. Formulae to express it in terms of the standard Radon transform (or vice versa) are given. We also discuss some extensions as applications.

A typical feature of spontaneous collapse models which aim at localizing wavefunctions in space is the violation of the principle of energy conservation. In the models proposed in the literature, the stochastic field which is responsible for the localization mechanism causes the momentum to behave like a Brownian motion, whose larger and larger fluctuations show up as a steady increase of the energy of the system. In spite of the fact that, in all situations, such an increase is small and practically undetectable, it is an undesirable feature that the energy of physical systems is not conserved but increases constantly in time, diverging for t → ∞. In this paper, we show that this property of collapse models can be modified:. we propose a model of spontaneous wavefunction collapse sharing all most important features of usual models but such that the energy of isolated systems reaches an asymptotic finite value instead of increasing with a steady rate.

A simple pseudo-Hamiltonian formulation is proposed for the linear inhomogeneous systems of ODEs. In contrast to the usual Hamiltonian mechanics, our approach is based on the use of a non-stationary Poisson bracket, i.e. the corresponding Poisson tensor is allowed to explicitly depend on time. Starting from this pseudo-Hamiltonian formulation, we develop a consistent deformation quantization procedure involving a non-stationary star-product ∗_t and an 'extended' operator of time derivative D_t = ∂_t + ⋅⋅⋅, differentiating the ∗_t-product. As in the usual case, the ∗_t-algebra of physical observables is shown to admit an essentially unique (time-dependent) trace functional Tr_t. Using these ingredients, we construct a complete and fully consistent quantum-mechanical description for any linear dynamical system with or without dissipation. The general quantization method is exemplified by the models of damped oscillator and radiating point charge.

We investigate N-extended supersymmetry in one-dimensional quantum mechanics on a circle with point singularities. For any integer n, N = 2n + 1 supercharges are explicitly constructed in terms of discrete transformations, and a class of singularities compatible with supersymmetry is clarified. In our formulation, the supersymmetry can be reduced to M-extended supersymmetry for any integer M < N. The degeneracy of the spectrum and spontaneous supersymmetry breaking are also studied.

The problem of the surface polariton existence in symmetry planes of non-magnetic biaxial crystals is studied theoretically. The plane interface of such a crystal and a semi-infinite isotropic medium is considered. With the use of the integral formalism formulated in our earlier work, the dispersion equation is derived for the polaritons under consideration. The existence conditions for the dispersion equation solutions are obtained in the form of algebraic inequalities for principal values of inverse dielectric permittivity tensors. If these conditions are satisfied, then excitation of surface waves is possible along the allowed propagation directions, which constitute sectors in the interface plane. Exact expressions are obtained that determine location of these sectors with respect to the symmetry axes of the crystal.

We study the asymptotics of the heat trace where P is an operator of Dirac type, where f is an auxiliary smooth smearing function which is used to localize the problem, and where we impose spectral boundary conditions. Using functorial techniques and special case calculations, the boundary part of the leading coefficients in the asymptotic expansion is found.

Computer-based modelling in ellipsoidal geometry can be useful in electro- encephalography and gastrography due to the resemblance of the human brain and stomach to an ellipsoid. Both theoretically and computationally, the bioelectric current dipole model is important for the study of electrical activity in these two organs. Computing the electric potential ϕ and electric field E due to a current dipole located in an ellipsoid requires truncated series expansions involving ellipsoidal harmonics , which are the products of Lamé functions. A theoretical model for this has appeared in the literature only for expansions of order 2 in ; however, this may be insufficient in forward electrogastrography, while an analogous situation is encountered in some geodetic problems where similar expansions are required. In this paper, we propose a generalized model for computing ϕ and E numerically using harmonic expansions of arbitrary order and degree. The implementation of such a procedure involves finding roots of Lamé polynomials of degree 5 or higher using an optimization technique for solving nonlinear systems of equations. This process can allow one to make a knowledgeable decision concerning the optimal expansion size required for the physical problem under investigation without compromising the overall accuracy of the computation.

For λϕ⁴ models, the introduction of a large field cutoff improves significantly the accuracy that can be reached with perturbative series but the calculation of the modified coefficients remains a challenging problem. We show that this problem can be solved numerically, and analytically in the limits of large and small field cutoffs, for the ground-state energy of the anharmonic oscillator. For the two lowest orders in λ, the approximate formulae obtained in the large field cutoff limit extend unexpectedly far in the low field cutoff region and there is a significant overlap with the small field cutoff approximation. For the higher orders, this is not the case; however the shape of the transition between the small field cutoff regime and the large field cutoff regime is approximately order independent.

In this paper we provide a complete description of the first integrals of the classical Einstein–Yang–Mills equations that can be described by formal series. As a corollary we also obtain a complete description of the analytic first integrals in a neighbourhood of the origin.

We investigate properties of plasma flow, governed by an exact reduction of the ideal magnetohydrodynamics equations which was obtained in our recent article. The reduction is called a singular vortex. It gives a description of a three-dimensional nonstationary plasma flow in terms of solution of reduced system of PDEs with two independent variables. In this paper we investigate symmetry properties of the reduced system and construct its invariant solutions. Also we give a description of the physical features of plasma motion, governed by the singular vortex. Namely, we prove that in the observed class of solutions the particles trajectories as well as magnetic field lines are flat curves. The position and orientation of the plane containing the particle's trajectory depends on the initial position of the particle. We describe the shape of trajectories and magnetic field lines for the cases of stationary and self-similar plasma flow.

There is a factor of 2 error in equation (61) of this paper. Correcting this error leads to minor changes in equations (62) and (63). Please see PDF for details.