On the existence of a gap in the energy spectrum of quantum systems

A theorem on the existence of a gap in the energy spectrum of quantum systems, the exact ground state of which is known explicitly, is proved. The theorem is applied to a three-dimensional Heisenberg spin-1/2 ferromagnet, with anisotropic nearest-neighbour interactions, and to an alternating Heisenberg antiferromagnet, with nearest- and next-nearest-neighbour interactions

Exact ground states for a class of linear quantum spin systems

The singlet pair state (S.P.S.) is shown to be the exact ground state for a class of linear quantum spin systems with anisotropic interactions

Competitive forms of symmetry breaking in linear antiferromagnetic systems

Two different forms of symmetry breaking are considered for linear antiferromagnetic systems (S = 1/2 ). Their relative stability is examined by considering small fluctuations in the harmonic oscillator approximation. Imaginary frequencies correspond with an unstable phase, and the ground state represents an absolute minimum of the total energy, including contributions from the zero-point fluctuations

Some exact excited states in a linear antiferromagnetic spin system

Exact expressions are derived for some excited states in a linear quantum spin system for which the exact ground state has been studied in the last decade

An analogue of the Magnus problem for associative algebras

We prove an analogue of the Magnus theorem for associative algebras without unity over arbitrary fields. Namely, if an algebra is given by n+k generators and k relations and has an n-element system of generators, then this algebra is a free algebra of rank n

The Majumdar-Ghosh chain. Twofold ground state and elementary excitations

Recently it was proved that the Majumdar-Ghosh chain with the Hamiltonian H=4 Sigma j=12NSj.Sj+1+2 Sigma j=12N Sj.Sj+2, Si+2N identical to Si, Si=1/2, has at least two ground states, in which the spins are arranged in nearest-neighbour singlet pairs. In this work it is shown that these two states are the only ground states. Besides, a rapidly converging variational method is given to determine the elementary excitation

Numerical Calculation of Bessel Functions

A new computational procedure is offered to provide simple, accurate and flexible methods for using modern computers to give numerical evaluations of the various Bessel functions. The Trapezoidal Rule, applied to suitable integral representations, may become the method of choice for evaluation of the many Special Functions of mathematical physics.Comment: 10 page

Calculation of Superdiffusion for the Chirikov-Taylor Model

It is widely known that the paradigmatic Chirikov-Taylor model presents enhanced diffusion for specific intervals of its stochasticity parameter due to islands of stability, which are elliptic orbits surrounding accelerator mode fixed points. In contrast with normal diffusion, its effect has never been analytically calculated. Here, we introduce a differential form for the Perron-Frobenius evolution operator in which normal diffusion and superdiffusion are treated separately through phases formed by angular wave numbers. The superdiffusion coefficient is then calculated analytically resulting in a Schloemilch series with an exponent $\beta=3/2$ for the divergences. Numerical simulations support our results.Comment: 4 pages, 2 figures (revised version

Breathers in the weakly coupled topological discrete sine-Gordon system

Existence of breather (spatially localized, time periodic, oscillatory) solutions of the topological discrete sine-Gordon (TDSG) system, in the regime of weak coupling, is proved. The novelty of this result is that, unlike the systems previously considered in studies of discrete breathers, the TDSG system does not decouple into independent oscillator units in the weak coupling limit. The results of a systematic numerical study of these breathers are presented, including breather initial profiles and a portrait of their domain of existence in the frequency-coupling parameter space. It is found that the breathers are uniformly qualitatively different from those found in conventional spatially discrete systems.Comment: 19 pages, 4 figures. Section 4 (numerical analysis) completely rewritte