Table of contents

We prove, using the coordinate Bethe ansatz, the exact solvability of a model of three particles whose point-like interactions are determined by the root system of . The statistics of the wavefunction are left unspecified. Using the properties of the Weyl group, we are also able to find Bethe equations. It is notable that the method relies on a certain generalized version of the well-known Yang–Baxter equation. A particular class of non-trivial solutions to this equation emerges naturally.

Various Euclidean, hyperbolic and elliptic analytic representations are introduced and relations among them are discussed. The Bargmann analytic representation in the complex plane is considered and its relation to other phase-space methods for the harmonic oscillator is reviewed. The general theory that relates the growth of analytic functions with the density of their zeros is applied to Bargmann functions and it leads to theorems on the completeness of sequences of Glauber coherent states. Two hyperbolic analytic representations in the unit disc, based on SU(1, 1) coherent states and also on phase states are introduced. A third analytic representation in the complex plane based on Barut–Girardello states is also considered and transformations which relate it to the other ones are studied. In the case of systems with finite-dimensional Hilbert space, an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced. The Berezin formalism in the Euclidean, hyperbolic and elliptic cases is discussed. Contour analytic representations in these three cases are also presented.

We introduce a class of one-dimensional lattice models in which a quantity, which may be thought of as an energy, is either transported from one site to a neighbouring one, or locally dissipated. Transport is controlled by a continuous bias parameter, q, which allows us to study symmetric as well as asymmetric cases. We derive sufficient conditions for the factorization of the N-body stationary distribution and give an explicit solution for the latter, before briefly discussing physically relevant situations.

We present an eigenvalue theory to study the stochastic dynamics of non-stationary time-periodic Markov processes. The analysis is carried out by solving an integral operator of the Fredholm type, i.e. considering complex-valued functions fulfilling the Kolmogorov compatibility condition. We show that the asymptotic behaviour of the stochastic process is characterized by the smaller time-scale associated with the spectrum of the Kolmogorov operator. The presence of time-periodic elements in the evolution equation for the semigroup leads to a Floquet analysis. The first non-trivial Kolmogorov's eigenvalue is interpreted from a physical point of view. This non-trivial characteristic time-scale strongly depends on the interplay between the stochastic behaviour of the process and the time-periodic structure of the Fokker–Planck equation for continuous processes, or the periodically modulated master equation for discrete Markov processes. We present pedagogical examples in a finite-dimensional vector space to calculate the Kolmogorov characteristic time-scale for discrete Markov processes.

We establish necessary and sufficient conditions for the existence of factorizable steady states of the generalized zero range process on a periodic or infinite lattice. This process allows transitions from a site i to a site (i + q) involving (a bounded number of) multiple particles with rates depending on the content of the site i, the direction q of movement, and the number of particles moving. We also show the sufficiency of a similar condition for the continuous time mass transport process, where the mass at each site and the amount transferred in each transition are continuous variables; we conjecture that this is also a necessary condition.

We derive an expression for the mean first passage time (MFPT) for the random walk with random step size on a one-dimensional linear lattice. Here both ends of the linear lattice are reflecting boundaries whereas the absorbing boundary is situated anywhere in between. When the size of the lattice is N and the random step size is k, we show that the MFPT associated with the escape of the random walker only through a specific point that is situated anywhere in the interval [0, N] at the limit as k → ∞ is which is independent of the initial position as well as the absorbing point associated with the random walker on the linear lattice under consideration. This result has potential applications in the analysis and understanding of the fundamental processes in molecular biology such as DNA–protein and DNA–probe interactions.

Entropic forces in models of Motzkin paths in a wedge geometry are considered as models of forces in polymers in confined geometries. A Motzkin path in the square lattice is a path from the origin to a point in the line Y = X while it never visits sites below this line, and it is constrained to give unit length steps only in the north and east directions and steps of length in the north-east direction. Motzkin path models may be generalized to ensembles of NE-oriented paths above the line Y = rX, where r > 0 is an irrational number. These are paths giving east, north and north-east steps from the origin in the square lattice, and confined to the r-wedge formed by the Y-axis and the line Y = rX. The generating function g_r of these paths is not known, but if r > 1, then I determine its radius of convergence to be and if r ∊ (0, 1), then t_r = 1/3. The entropic force the path exerts on the line Y = rX may be computed from this. An asymptotic expression for the force for large values of r is given by In terms of the vertex angle α of the r-wedge, the moment of the force about the origin has leading terms as α → 0⁺ and F(α) = 0 if α ∊ [π/4, π/2]. Moreover, numerical integration of the force shows that the total work done by closing the wedge is 1.085 07... lattice units. An alternative ensemble of NE-oriented paths may be defined by slightly changing the generating function g_r. In this model, it is possible to determine closed-form expressions for the limiting free energy and the force. The leading term in an asymptotic expansions for this force agrees with the leading term in the asymptotic expansion of the above model, and the subleading term only differs by a factor of 2.

We compute the fractal dimensions of the 'hulls' or external boundary and the boundaries of the internal cavities in several deterministic as well as random fractal structures. Our conclusion is that the two fractal dimensions are in fact identical. The deterministic fractals we study are Sierpinski carpets (SC) in a two-dimensional space and the random fractals are percolation clusters at criticality. As an intermediate case, we present results on some randomized SC. In the random structures, statistics of the area and perimeters of all internal cavities or holes are taken and the fractal dimension of the hull borderline is computed. Two different definitions of the borderline are used, considering nearest neighbours as well as nearest and second nearest neighbours as connected. The conclusion is valid for both cases.

We consider the problem of existence of certain symmetrical solutions of the Stokes equation on a three-dimensional manifold M with a general metric possessing symmetry. These solutions correspond to unidirectional flows. We have been able to determine necessary and sufficient conditions for their existence. Symmetric unidirectional flows are fundamental for deducing the so-called Darcy's law, which is the law governing fluid flow in a Hele–Shaw cell embedded in the environment M. Our main interest is to depart from the usual, flat background environment and consider the possibility of an environment of arbitrary constant curvature K in which a cell is embedded. We generalize Darcy's law for particular models of such spaces obtained from with a conformal metric. We employ the calculus of differential forms for a simpler and more elegant approach to the problems discussed.

We calculate the eigenvalue ρ of the multiplication mapping R on the Cayley–Dickson algebra . If the element in is composed of a pair of alternative elements in , half the eigenvectors of R in are still eigenvectors in the subspace which is isomorphic to . The invariant under the reciprocal transformation plays a fundamental role in simplifying the functional form of ρ. If some physical field can be identified with the eigenspace of R, with an injective map from the field to a scalar quantity (such as a mass) m, then there is a one-to-one map π:m ↦ ρ. As an example, the electro-weak gauge field can be regarded as the eigenspace of R, where π implies that the W-boson mass is less than the Z-boson mass, as in the standard model.

The problem of estimating the spectrum of a density matrix is considered. Other problems, such as bipartite pure state entanglement, can be reduced to spectrum estimation. A local operations and classical communication (LOCC) measurement strategy is shown which is asymptotically optimal. This means that, for a very large number of copies, it becomes unnecessary to perform collective measurements which should be more difficult to implement in practice.

A non-Hermitian Hamiltonian that has an unbroken symmetry can be converted by means of a similarity transformation to a physically equivalent Hermitian Hamiltonian. This raises the following question: in which form of the quantum theory, the non-Hermitian or the Hermitian one, is it easier to perform calculations? This paper compares both forms of a non-Hermitian ix³ quantum-mechanical Hamiltonian and demonstrates that it is much harder to perform calculations in the Hermitian theory because the perturbation series for the Hermitian Hamiltonian is constructed from divergent Feynman graphs. For the Hermitian version of the theory, dimensional continuation is used to regulate the divergent graphs that contribute to the ground-state energy and the one-point Green's function. The results that are obtained are identical to those found much more simply and without divergences in the non-Hermitian -symmetric Hamiltonian. The contribution to the ground-state energy of the Hermitian version of the theory involves graphs with overlapping divergences, and these graphs are extremely difficult to regulate. In contrast, the graphs for the non-Hermitian version of the theory are finite to all orders and they are very easy to evaluate.

Weyl's unitary operators for displacement in position and momentum commute with one another if the product of the elementary displacements equals Planck's constant. Then, their common eigenstates constitute the Zak basis, with each state specified by two phase parameters. Accordingly, the transformation function from the position basis to the Zak basis maps the Hilbert space on the line onto the Hilbert space on the torus. This mapping is one to one provided that the Zak basis states are periodic functions of their phase parameters, but then the mapping cannot be continuous on the whole torus. With the periodicity of the Zak basis enforced, the basis has a double Fourier series. The Fourier coefficients identify a discrete basis which complements the periodic Zak basis to form a pair of mutually unbiased bases. The discrete basis states are the common eigenstates of the two complementary partners to the two unitary displacement operators. These partner operators are of angular-momentum type, with integer eigenvalues, and generate the fundamental rotations of the torus. Conversely, the displacement operators are the ladder operators for their partners. For each consistent phase convention for the periodic Zak basis, and thus for the line-onto-torus mapping, there is a corresponding discrete Zak basis and a corresponding pair of partner operators. Examples of particular interest are the conventions that give a continuous mapping in one phase parameter or are symmetric in both phase parameters. The latter emphasizes the Heisenberg–Weyl symmetry between position and momentum. We discuss briefly the relation between the Zak bases and Aharonov's modular operators. Finally, as an application of the Zak operators for the torus, we mention how they can be used to associate with the single degree of freedom of the line a pair of genuine qubits that are potentially entangled.

We investigate the convergence properties of a perturbation method proposed some time ago and reveal some of its most interesting features. Anharmonic oscillators in the strong-coupling limit prove to be appropriate illustrative examples and a suitable benchmark.

We reformulate the Lanczos algorithm for quantum wavefunction propagation in terms of the variational principle. By including some basis states of previous time steps in the variational subspace, the resultant accuracy increases by several orders. Numerical errors of the alternative method accumulate much slower than that of the original Lanczos method. There is almost no extra numeric cost for the gaining of the accuracy, i.e., the accuracy increase needs no extra operations of the Hamiltonian acting on state vectors, which are the major numeric cost for wavefunction propagation. A wave packet moving in a two-dimensional Hénon–Heiles model serves as an illustration. This method is suitable for small time step propagation of quantum wavefunctions in large-scale time-dependent calculations where the operations of the Hamiltonian acting on state vectors are expensive.

The general conditions are discussed which quantum state purification protocols have to fulfil in order to be capable of purifying Bell-diagonal qubit-pair states, provided they consist of steps that map Bell-diagonal states to Bell-diagonal states and they finally apply a suitably chosen Calderbank–Shor–Steane code to the outcome of such steps. As a main result a condition on asymptotic correctability is presented, which relates this problem to the magnitude of a characteristic exponent governing the relation between bit and phase errors under the purification steps. This condition allows a straightforward determination of maximum tolerable bit-error rates of quantum key distribution protocols whose security analysis can be reduced to the purification of Bell-diagonal states.

In previous work, it was shown numerically that under successive scattering events, a collection of particles with Gaussian wavefunctions retains the Gaussian property, with the spread of the Gaussian ('Δx') tending to a value inversely proportional to the square root of each particle's mass. We prove this convergence in all dimensions ⩾3.

The neat formulation that describes the gauge interactions associated with internal symmetries is extended to the case of a simple, yet non-trivial, symmetry group structure which mixes gravity and electromagnetism by associating a gauge symmetry with a central extension of the Poincaré group.

We present an analysis of the gonihedric loop model, a reformulation of the two-dimensional gonihedric spin model, using two different techniques. First, the usual regular lattice statistical physics problem is mapped onto a height model and studied analytically. Second, the gravitational version of this loop model is studied via matrix models techniques. Both methods lead to the conclusion that the model has c_matter = 0 for all values of the parameters of the model. In this way it is possible to understand the absence of a continuous transition.

We derive a formula describing the evolution of tip splittings of Saffman–Taylor fingers in a Hele–Shaw cell, at zero surface tension.