Table of contents

We consider the spectrum of an N-particle Hamiltonian H = ∑_i(1/2m_i)Δ_i + ∑_i<jV_ij(-_ji) with translation-invariant pair interactions V_ij. H acts in L²(R^3N). Letting σ denote the spectrum we obtain the lower bound inf σ(H)⩾ ∑_i<jinf σ(h_ij') where h_ij' = -(1/2µ_ij')Δ + V_ij(), 1⩽i<j⩽N is the single-particle relative coordinate Hamiltonian with reduced mass µ_ij' = (N-1)µ_ij, µ_ij = m_im_j(m_i + m_j)^-1 acting in L²(R³). In particular, if σ(h_ij')⊂[0,∞) (for example, weak pair interactions) for all i,j then H has no negative energy spectrum. For example, if each V_ij() is in L^3/2(R³) and sufficiently small it is known that σ(h_ij')⊂[0,∞).

We study a continuum interfacial Hamiltonian model of fluid adsorption in a (1+1)-dimensional wedge geometry, which is known to exhibit a filling transition when the contact angle θ_π = α, with α the wedge angle. We extend the transfer matrix analysis of the model to calculate the interfacial height probability distribution function P(l;x), for arbitrary positions x along the wedge. The asymptotics of this function reveal a fluctuation-induced disorder point (non-thermodynamic singularity) that occurs prior to filling when θ_π = 2α, where there is a change of length scales determining the decay of P(l;x).

We consider the one-dimensional classical Ising model in a symmetric dichotomous random field. The problem is reduced to a random iterated function system (RIFS) for an effective field. The D_q-spectrum of the invariant measure of this effective field exhibits a sharp drop of all D_q with q<0 at some critical strength of the random field. We introduce the concept of orbits, which naturally group the points of the support of the invariant measure. We then show that the pointwise dimension at all points of an orbit has the same value and calculate it for a class of periodic orbits and their so-called offshoots as well as for generic orbits in the non-overlapping case. The sharp drop in the D_q-spectrum is analytically explained by a drastic change of the scaling properties of the measure near the points of a certain periodic orbit at a critical strength of the random field, which is explicitly given. A similar drastic change near the points of a special family of periodic orbits explains a second, hitherto unnoticed transition in the D_q-spectrum. As it turns out, a decisive role in this mechanism is played by a specific offshoot. We furthermore give rigorous upper and/or lower bounds on all D_q in a wide parameter range. In most cases the numerically obtained D_q coincide with either the upper or the lower bound. The results in this paper are relevant for the understanding of RIFSs in the case of moderate overlap, in which periodic orbits with weak singularity can play a decisive role.

The correlations between topological and metric properties of fractal tessellations are analysed. To this end, Sierpinski cellular structures are constructed for different geometries related to Sierpinski gaskets and to the Apollonian packing of discs. For these geometries, the properties of the distribution of the cells' areas and topologies can be derived analytically. In all cases, an algebraic increase of the cell's average area with its number of neighbours is obtained. This property, unknown from natural cellular structures, confirms previous observations in numerical studies of Voronoi tessellations generated by fractal point sets. In addition, a simple rigorous scaling resp. multiscaling properties relating the shapes and the sizes of the cells are found.

We study the ground state Ψ₀ = ∑a_X|X⟩ of N hard-core bosons on a finite lattice in configuration space, X = {x₁,...,x_N}. All a_X being positive, the ratios a_X/∑a_Y can be interpreted as probabilities P_a(X). Let E₀ denote the energy of the ground state and |∂X| the number of nearest-neighbour particle-hole pairs in the configuration X. We prove the concentration of P_a onto X with |∂X| in a (|E₀|)^1/2-neighbourhood of |E₀|, show that the average of a_X over configurations with |∂X| = n increases exponentially with n, discuss fluctuations about this average, derive upper and lower bounds on E₀ and give an argument for off-diagonal long-range order in the ground state.

We obtain bounds on the complex eigenvalues of non-self-adjoint Schrödinger operators with complex potentials, and also on the complex resonances of self-adjoint Schrödinger operators. Our bounds are compared with numerical results, and are seen to provide useful information.

In this paper a non-relativistic particle moving on a hypersurface in a curved space and the multidimensional rotator are investigated using the Hamilton-Jacobi formalism. The equivalence with the Dirac Hamiltonian formalism is demonstrated in both Cartesian and curvilinear coordinates. The energy spectrum of the multidimensional rotator is equal to that of a pure Laplace-Beltrami operator with no additional constant arising from the curvature of the sphere.

The Bose condensate model is used to elucidate the motion of the electron bubble in superfluids. An asymptotic expansion is developed for steady subcritical flow. Numerical integration of the coupled nonlinear Schrödinger equations, that describe the evolution of the wavefunctions of the Bose condensate and the impurity, is used to study the nucleation and capture of vortex rings. Because an electron bubble is made oblate by its motion relative to the condensate, the critical velocity for the vortex nucleation is reduced by about 20%, in agreement with experiments.

Quantum tomography is the process of reconstructing the ensemble average of an arbitrary operator (observable or not, including the density matrix), which may not be directly accessible by feasible detection schemes, starting from the measurement of a complete set of observables i.e. a quorum. The measurement of a quorum thus represents a complete characterization of the quantum state. The operator expression in terms of a quorum corresponds to an expansion on an irreducible set of operators in the Liouville space. We give two general characterizations of these sets, and show that all the known quantum tomographies can be described in this framework. New operatorial resolutions are also given that may be used in novel reconstruction schemes.

A combinatorial formula of Rota and Stein is taken to perform Wick reordering in quantum field theory. Wick's theorem becomes a Hopf algebraic identity called Cliffordization. The combinatorial method relying on Hopf algebras is highly efficient in computations and yields closed algebraic expressions.

The Karpman-Kaup equation is a generic limit model for one-dimensional long-wave short-wave resonant interaction within the Benney criterion. We show that (1) the input short-wave fields generate solitons, (2) the boundary inputs allow one to drive the soliton, (3) the spectral transform tool can be adapted to solve (linearize) the finite-interval problem and (4) the explicit solution provided by the semi-infinite interval limit produces a very accurate description of the finite-interval case. The argumentation is based on the solution of this equation by the inverse spectral transform and the results are obtained by numerical simulation of both the original system and its spectral transform. The main issue is the demonstration that the spectral transform is a quite practical and efficient tool and that it has in this case a simple and explicit expression in terms of the input and output short-wave field values.

We use a non-Lagrangian method to express the generalized nonlinear Schrödinger equation, for pulse propagation in optical fibres, in terms of the pulse parameters, called collective variables, such as the pulse width, amplitude, chirp and frequency. The collective variable equations of motion, which include the important effects due to fibre losses, third-order dispersion, stimulated Raman scattering and self-steepening are derived using a direct averaging method without the help of any Lagrangian.

We apply the method of group foliation to the complex Monge-Ampère equation ({CMA}₂) with the goal of establishing a regular framework for finding its non-invariant solutions. We employ the infinite symmetry subgroup of the equation, the group of unimodular biholomorphisms, to produce a foliation of the solution space into leaves which are orbits of solutions with respect to the symmetry group. Accordingly, {CMA}₂ is split into an automorphic system and a resolvent system which we derive in this paper. This is an intricate system and here we make no attempt to solve it in order to obtain non-invariant solutions.

We obtain all differential invariants up to third order for the group of unimodular biholomorphisms and, in particular, all the basis differential invariants. We construct the operators of invariant differentiation from which all higher differential invariants can be obtained. Consequently, we are able to write down all independent partial differential equations with one real unknown and two complex independent variables which keep the same infinite symmetry subgroup as {CMA}₂. We prove explicitly that applying operators of invariant differentiation to third-order invariants we obtain all fourth-order invariants. At this level we have all the information which is necessary and sufficient for group foliation.

We propose a new approach in the method of group foliation which is based on the commutator algebra of operators of invariant differentiation. The resolving equations are obtained by applying this algebra to differential invariants with the status of independent variables. Furthermore, this algebra together with Jacobi identities provides the commutator representation of the resolvent system. This proves to be the simplest and most natural way of arriving at the resolving equations.

We report for the first time exact solutions of a completely integrable nonlinear lattice system for which the dynamical variables satisfy a q-deformed Lie algebra - the Lie-Poisson algebra su_q(2). The system considered is a q-deformed lattice for which in the continuum limit the equations of motion become the envelope Maxwell-Bloch (or SIT) equations describing the resonant interaction of light with a nonlinear dielectric. Thus the N-soliton solutions we report here are the natural q-deformations, necessary for a lattice, of the well known multi-soliton and breather solutions of self-induced transparency (SIT). The method we use to find these solutions is a generalization of the Darboux-Bäcklund dressing method. The extension of these results to quantum solitons is sketched.

Stationary geodesics of left-invariant Riemannian metrics on Lie groups were introduced by Arnold as those geodesics which are also orbits of one-parameter groups of left-translations. The existence of infinitely many stationary geodesics in the case of compact semi-simple Lie groups has recently been established by Szenthe J. Stationary geodesics of left-invariant Lagrangians on Lie groups are studied and the existence of infinitely many such geodesics on compact semi-simple Lie groups is established below.

We derive an algorithm for calculating Lie point symmetries of systems of stochastic ordinary differential equations (SODEs) of any order. From this algorithm, the following facts emerge. Symmetries of a SODE do not in general form a Lie algebra. The determining equations for Ito's equation are in general stochastic linear partial differential equations whereas for SODEs of order n⩾2 the determining equations are linear deterministic partial differential equations that form an overdetemined system which is solvable by classical methods.

For scalar second-order SODEs, we provide a complete classification of equations admitting finite-dimensional symmetry Lie algebras. This classification is applied to the integration of scalar second-order SODEs: in general a SODE admitting a two-dimensional symmetry algebra is not integrable by quadratures, although it is reducible to a homogeneous Ito equation. In particular, a scalar second-order SODE admitting a two-dimensional symmetry algebra with connected operators is linearizable. We also characterize integrable scalar second-order SODEs admitting three-dimensional symmetry algebras. Finally we show that a SODE can admit maximally a zero-, one-, two-, three- or four-dimensional Lie algebra.

A new (2+1)-dimensional integrable soliton equation is proposed, which has a close connection with the Levi soliton hierarchy. Through the nonlinearization of the Levi eigenvalue problems, we obtain a finite-dimensional integrable system. The Abel-Jacobi coordinates are constructed to straighten out the Hamiltonian flows, by which the solutions of both the 1 + 1 and 2 + 1 Levi equations are obtained through linear superpositions. An inversion procedure gives the quasi-periodic solution in the original coordinates in terms of the Riemann theta functions.