Table of contents

Jordan–Wigner-type transformations connecting the spin-3/2 operators and two types of fermion are derived. The general condition of fermionizability of spins is obtained. Discordances in the results of a previous attempt to generalize the Jordan–Wigner transformation for all spins (Batista and Ortiz 2001 Phys. Rev. Lett.86 1082) are pointed out. After introducing clarity, a new interpretation is given for these results.

In this letter we will extend the analysis given by Zamolodchikov for the scaling Yang–Lee model on the sphere to the Ising model in a magnetic field. A numerical study of the partition function and of the vacuum expectation values is done by using the truncated conformal space approach. Our results strongly suggest that the partition function is an entire function of the coupling constant.

We formulate noncommutative qauntum field theory in terms of fields defined as mean value over coherent states of the noncommutative plane. No ∗-product is needed in this formulation and noncommutativity is carried by a modified Fourier transform of fields. As a result the theory is UV finite and the cutoff is provided by the noncommutative parameter θ.

We investigate the motion of a tagged spin in a ferromagnetic Ising chain evolving under Kawasaki dynamics. At equilibrium, the displacement is Gaussian, with a variance increasing as At^1/2. The temperature dependence of the prefactor A is derived exactly. At low temperature, where the static correlation length ξ is large, the mean square displacement increases as (t/ξ²)^2/3 in the coarsening regime, i.e., as a finite fraction of the mean square domain length. The case of totally asymmetric dynamics, where (+) (resp. (−)) spins move only to the right (resp. to the left), is also considered. In the steady state, the displacement variance increases as Bt^2/3. The temperature dependence of the prefactor B is derived exactly, using the Kardar–Parisi–Zhang theory. At low temperature, the displacement variance increases as t/ξ² in the coarsening regime, again proportionally to the mean square domain length.

We consider the resolvent of a system of first-order differential operators with a regular singularity, admitting a family of self-adjoint extensions. We find that the asymptotic expansion for the resolvent in the general case presents powers of λ which depend on the singularity, and can take even irrational values. The consequences for the pole structure of the corresponding ζ- and η-functions are also discussed.

We use the Whittaker–Shannon sampling theorem to show that the eigenvalue problem for the sinc-kernel is equivalent to a discrete eigenvalue problem. The well-known eigenfunctions, namely, the prolate spheroidal wavefunctions, their corresponding eigenvalues and the orthogonality and completeness properties are determined without invoking the prolate spheroidal differential equation. This analysis based on the sampling theorem may be used for calculating the eigenvalues and eigenfunctions of bandlimited kernels in general as we illustrate with an additional example of the sinc²-kernel.

We extend the Mason–Newman Lax pair for the elliptic complex Monge–Ampère equation so that this equation itself emerges as an algebraic consequence. We regard the function in the extended Lax equations as a complex potential. Their differential compatibility condition coincides with the determining equation for the symmetries of the complex Monge–Ampère equation. We shall identify the real and imaginary parts of the potential, which we call partner symmetries, with the translational and dilatational symmetry characteristics, respectively. Then we choose the dilatational symmetry characteristic as the new unknown replacing the Kähler potential. This directly leads to a Legendre transformation. Studying the integrability conditions of the Legendre-transformed system we arrive at a set of linear equations satisfied by a single real potential. This enables us to construct non-invariant solutions of the Legendre transform of the complex Monge–Ampère equation. Using these solutions we obtained explicit Legendre-transformed hyper-Kähler metrics with a anti-self-dual Riemann curvature 2-form that admit no Killing vectors. They satisfy the Einstein field equations with Euclidean signature. We give the detailed derivation of the solution announced earlier and present a new solution with an added parameter. We compare our method of partner symmetries for finding non-invariant solutions to that of Dunajski and Mason who use 'hidden' symmetries for the same purpose.

We study stationary solutions of the nonlinear Schrödinger equation in the presence of small but non-zero third-order dispersion (TOD). Using a singular perturbation theory around the ideal soliton we calculate these solutions up to the second order in the TOD coefficient. The existence and linear stability of the stationary solutions is proved for any finite order of the perturbation theory. The results obtained by our numerical simulations of the nonlinear Schrödinger equation are in very good agreement with theory. The significance of these results for fibre optic communication systems is discussed.

The potentials for a one-dimensional Schrödinger equation that are displaced along the x-axis under second-order Darboux transformations, called 2-SUSY invariant, are characterized in terms of a differential–difference equation. The solutions of the Schrödinger equation with such potentials are given analytically for any value of the energy. The method is illustrated by a two-soliton potential. It is proved that a particular case of the periodic Lamé–Ince potential is 2-SUSY invariant. Both Bloch solutions of the corresponding Schrödinger equation are found for any value of the energy. A simple analytic expression for a family of two-gap potentials is derived.

A method of generating some quantum mechanical exactly solvable potentials (ESPs), using the properties of classical orthogonal polynomials (COPs), is presented. It is illustrated using hypergeometric polynomial. Utilization of other COPs to get more ESPs is indicated.

We compute the volume of the (N² − 1)-dimensional set _N of density matrices of size N with respect to the Bures measure and show that it is equal to that of an (N² − 1)-dimensional hyper-hemisphere of radius 1/2. For N = 2 we obtain the volume of the Uhlmann hemisphere, ½S³ ⊂ ⁴. We find also the area of the boundary of the set _N and obtain analogous results for the smaller set of all real density matrices. An explicit formula for the Bures–Hall normalization constants is derived for an arbitrary N.

The operator-Schmidt decomposition is useful in quantum information theory for quantifying the nonlocality of bipartite unitary operations. We construct a family of unitary operators on ⁿ ⊗ ⁿ whose operator-Schmidt decompositions are computed using the discrete Fourier transform. As a corollary, we produce unitaries on ³ ⊗ ³ with operator-Schmidt number S for every S ∊ {1, ..., 9}. This corollary was unexpected, since it contradicted reasonable conjectures of Nielsen et al (2003 Phys. Rev. A 67 052301) based on intuition from a striking result in the two-qubit case. By the results of Dür et al (2002 Phys. Rev. Lett.89 057901), who also considered the two-qubit case, our result implies that there are nine equivalence classes of unitaries on ³ ⊗ ³ which are probabilistically interconvertible by (stochastic) local operations and classical communication. As another corollary, a prescription is produced for constructing maximally-entangled unitaries from biunimodular functions. Reversing tact, we state a generalized operator-Schmidt decomposition of the quantum Fourier transform considered as an operator ^M₁ ⊗ ^M₂ → ^N₁ ⊗ ^N₂, with M₁M₂ = N₁N₂. This decomposition shows (by Nielsen's bound) that the communication cost of the QFT remains maximal when a net transfer of qudits is permitted. In an appendix, a canonical procedure is given for removing basis-dependence for results and proofs depending on the 'magic basis' introduced in S Hill and W Wootters (1997 Entanglement of a pair of quantum bits Phys Rev. Lett.78 5022–5).

We compute the volume of the convex (N² − 1)-dimensional set _N of density matrices of size N with respect to the Hilbert–Schmidt measure. The hyper-area of the boundary of this set is also found and its ratio to the volume provides information about the structure of _N. Similar investigations are also performed for the smaller set of all real density matrices. As an intermediate step, we analyse volumes of the unitary and orthogonal groups and of the flag manifolds.

Axisymmetric transient solutions to the inhomogeneous telegraph equation are constructed in terms of spherical harmonics. Explicit solutions of the initial-value problem are derived in the spacetime domain by means of the Smirnov method of incomplete separation of variables and the Riemann formula. The corresponding Riemann function is constructed with the help of the Olevsky theorem. Solutions for some source distributions on a sphere expanding with a velocity greater than the wavefront velocity are obtained. This allows an analogous solution in the case of a circle belonging to a sphere expanding with the wavefront velocity to be written at once. Application of the scalar solution to a description of electromagnetic waves is also discussed.