Table of contents

This is a call for contributions to a special issue ofJournal of Physics A: Mathematical and General entitled `Singular Interactions in Quantum Mechanics: Solvable Models'. This issue should be a repository for high quality original work. We are interested in having the topic interpreted broadly, that is, to include contributions dealing with point-interaction models, one- and many-body, quantum graphs, including graph-like structures coupling different dimensions, interactions supported by curves, manifolds, and more complicated sets, random and nonlinear couplings, etc., as well as approximations helping us to understand the meaning of singular couplings and applications of such models on different parts of quantum mechanics. We believe that when the second printing of the `bible' of the field, the book Solvable Models in Quantum Mechanics by S Albeverio, F Gesztesy, the late R Høegh-Krohn and H Holden, appears it is the right moment to review new developments in this area, with the hope of stimulating further development of these extremely useful techniques.

The Editorial Board has invited G Dell'Antonio, P Exner and V Geyler to serve as Guest Editors for the special issue. Their criteria for acceptance of contributions are as follows:

• The subject of the paper should relate to singular interactions in quantum mechanics in the sense described above.

• Contributions will be refereed and processed according to the usual procedure of the journal.

• Papers should be original; reviews of a work published elsewhere will not be accepted.

The guidelines for the preparation of contributions are as follows:

• The DEADLINE for submission of contributions is 31 October 2004. This deadline will allow the special issue to appear in about April 2005.

• There is a nominal page limit of 15 printed pages (approximately 9000 words) per contribution. Papers exceeding these limits may be accepted at the discretion of the Guest Editors. Further advice on publishing your work in Journal of Physics A: Mathematical and General may be found at www.iop.org/Journals/jphysa.

• Contributions to the Special Issue should if possible be submitted electronically by web upload at {www.iop.org/Journals/jphysa or by e-mail to jphysa@iop.org, quoting `JPhysA Special Issue-Quantum Mechanics: Solvable Models'. Submissions should ideally be in standard LaTeX form; we are, however, able to accept most formats including Microsoft Word. Please see the web site for further information on electronic submissions.

• Authors unable to submit electronically may send hard copy contributions to: Publishing Administrators, Journal of Physics A, Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK, enclosing the electronic code on floppy disk if available and quoting `JPhysA Special Issue-Quantum Mechanics: Solvable Models'.

• All contributions should be accompanied by a read-me file or covering letter giving the postal and e-mail addresses for correspondence. The Publishing Office should be notified of any subsequent change of address.

This special issue will be published in the paper and online version of the journal. The corresponding author of each contribution will receive a complimentary copy of the issue.

G Dell'Antonio, P Exner and V Geyler

Guest Editors

For the first time, exact analytical solutions of electronic wavefunctions in quantum ring-like structures are found using a symmetry-adapted coordinate system. In the case of an infinite potential barrier, it is related to a more general problem of finding resonance modes in a ring-like cavity. The solutions in the parabolic coordinate system are Bessel or confluent hypergeometric functions for quantum dot, ring and wire including objects with flat bases. The continuous evolution from the 0D quantum dot system to the perfect 1D quantum ring system is analysed. The electronic properties (energy levels, wavefunctions and density of states) are continuously followed during the dot to ring transformation until the limit of a perfect 1D quantum ring is reached. The creation of a ring structure by changing the material composition in a dot centre leads to a sharp variation in the energy levels of totally symmetric states. This is a consequence of the immediate build-up of a potential barrier energy in the centre of the ring. Finally the use of the structure of electronic levels to identify the signature of self-organized quantum rings when compared to quantum dots is proposed.

We consider the problem of finite sample corrections for entropy estimation. New estimates of the Shannon entropy are proposed and their systematic error (the bias) is computed analytically. We find that our results cover correction formulae of current entropy estimates recently discussed in the literature. The trade-off between bias reduction and the increase of the corresponding statistical error is analysed.

We analyse the effects of the impermeability constraint on the equilibrium shapes of a three-dimensional vesicle hosting a rigid inclusion. A given alteration of the inclusion and/or vesicle parameters leads to shape modifications of different orders of magnitude, when applied to permeable or impermeable vesicles. Moreover, the enclosed-volume constraint wrecks the uniqueness of stationary equilibrium shapes, and gives rise to pear-shaped or stomatocyte-like vesicles.

A self-avoiding walk adsorbing on a line in the square lattice, and on a plane in the cubic lattice, is studied numerically as a model of an adsorbing polymer in dilute solution. The walk is simulated by a multiple Markov chain Monte Carlo implementation of the pivot algorithm for self-avoiding walks. Vertices in the walk that are visits in the adsorbing line or plane are weighted by e^β. The critical value of β, where the walk adsorbs on the adsorbing line or adsorbing plane, is determined by considering energy ratios and approximations to the free energy. We determine that the critical values of β are In addition, the value of the crossover exponent is determined: Metric quantities, including the mean square radius of gyration, are also considered, as well as rescaling of the specific heat and free energy, as the critical point is approached.

We use very efficient algorithms to calculate low-density series for bond and site percolation on the directed triangular, honeycomb, kagomé and (4.8²) lattices. Analysis of the series yields accurate estimates of the critical point p_c and various critical exponents. The exponent estimates differ only in the 5th digit, thus providing strong numerical evidence for the expected universality of the critical exponents for directed percolation problems. In addition we also study the non-physical singularities of the series.

We study the zero-temperature Ising chain evolving according to the Swendsen–Wang dynamics. We determine analytically the domain length distribution and various 'historical' characteristics, e.g. the density of unreacted domains is shown to scale with the average domain length as ⟨l⟩^−δ with δ = 3/2 (for the q-state Potts model, δ = 1 + q⁻¹). We also compute the domain length distribution for the Ising chain endowed with the zero-temperature Wolff dynamics.

We present an approximate analytical solution of the Cahn–Hilliard equation describing the coalescence during a first-order phase transition. We have identified all the intermediate profiles, stationary solutions of the noiseless Cahn–Hilliard equation. Using properties of the soliton lattices, periodic solutions of the Ginzburg–Landau equation, we have constructed a family of ansätze describing continuously the process of destabilization and period doubling predicted in Langer's self-similar scenario (Langer 1971 Ann. Phys., NY65 53).

A simple semiclassical approach, based on the investigation of anti-Stokes line topology, is presented for calculating Regge poles for nonsingular (Thomas–Fermi type) potentials, namely potentials with singularities at the origin weaker than order −2. The anti-Stokes lines for Thomas–Fermi potentials have a more complicated structure than those of singular potentials and require careful application of complex analysis. The explicit solution of the Bohr–Sommerfeld quantization condition is used to obtain approximate Regge poles. We introduce and employ three hypotheses to obtain several terms of the Regge pole approximation.

Please note that the pdf of this article was replaced with a corrected version on 29 June 2004. Minor changes have been made to pages 6950–6953.

We give a geometrical characterization of λ-prolongations of vector fields, and hence of λ-symmetries of ODEs. This allows an extension to the case of PDEs and systems of PDEs; in this context the central object is a horizontal 1-form μ, and we speak of μ-prolongations of vector fields and μ-symmetries of PDEs. We show that these are as good as standard symmetries in providing symmetry reduction of PDEs and systems, and explicit invariant solutions.

We present a coherent state quantization of the dynamics of a relativistic test particle on a one-sheet hyperboloid embedded in a three-dimensional Minkowski space. The group SO₀(1, 2) is considered to be the symmetry group of the system. Our procedure relies on the choice of coherent states of the motion on a circle. The coherent state realization of the principal series representation of SO₀(1, 2) seems to be a new result.

Equations of motion are derived for a fractional dimensional system of n-spatial coordinates to be used as an effective description of anisotropic and confined systems. An existing measure theoretic approach is extended to multiple variables and different degrees of confinement in orthogonal directions and comparisons are made with the analytic continuation of Gaussian integrals. This is applied to the variational principle, and equations of motion for a field described by a Lagrange density are found. A specific example is looked at in Schrödinger wave mechanics, particularly in three-coordinate systems.

We study the steady two-dimensional boundary-layer equations in the flat and axisymmetric case to show that some similarity reductions already found in the literature with ad hoc new methods of reduction are indeed invariant solutions under the action of non-classical symmetries (as introduced by Bluman and Cole). Moreover, we show that the celebrated von Mises transformation that reduces the two-dimensional flat boundary-layer equations to a second-order evolution equation is indeed a Bäcklund transformation related to a non-classical symmetry.

The transformation of the path integral measure under the reduction procedure in dynamical systems with a symmetry is considered. Investigation is carried out in the case of the Wiener-type path integrals that are used for description of the diffusion on a smooth compact Riemannian manifold with the given free isometric action of the compact semisimple unimodular Lie group. The transformation of the path integral, which factorizes the path integral measure, is based on the application of the optimal nonlinear filtering equation from the stochastic theory. The integral relation between the kernels of the original and reduced semigroups is obtained.

In this work we show how to define the action of a scalar field such that the Robin boundary condition is implemented dynamically, i.e. as a consequence of the stationary action principle. We discuss the quantization of that system via functional integration. Using this formalism, we derive an expression for the Casimir energy of a massless scalar field under Robin boundary conditions on a pair of parallel plates, characterized by constants c₁ and c₂. Some special cases are discussed; in particular, we show that for some values of c₁ and c₂ the Casimir energy as a function of the distance between the plates presents a minimum. We also discuss the renormalization at one-loop order of the two-point Green function in the λϕ⁴ theory subject to the Robin boundary condition on a plate.

In this paper we show how the method of [4] for the calculation of two-point functions in (d + 1)-dimensional AdS space can be simplified. This results in an algorithm for the evaluation of the two-point functions as linear combinations of Legendre functions of the second kind. This algorithm can be easily implemented on a computer. For the sake of illustration, we displayed the results for the case of symmetric traceless tensor fields with rank up to l = 4.