Spectrum of the energy operator of two-magnon systems in the isotropic Heisenberg ferromagnet model with impurity

We consider a two-magnon system in the isotropic Heisenberg ferromagnet model with impurity on a ν-dimensional lattice ℤ^ν. We establish that the essential spectrum of the system consists of the union of at most four intervals. We obtain the lower and upper estimates for the number of three-particle bound states of the system.     

Spectrum of the energy operator of a two-magnon system in the three-dimensional isotropic Heisenberg ferromagnet model with impurity

We consider the energy operator of two-magnon systems in the three-dimensional isotropic Heisenberg ferromagnet model with impurity and with the nearest-neighbor interaction. We investigate the structure of the essential spectrum and discrete spectrum of the system on a three-dimensional lattice. We show that the essential spectrum consists of the union of at most four segments and that the discrete spectrum is finite at the edge of the essential spectrum.     

Periodic weak solutions for a classical one-dimensional isotropic biquadratic Heisenberg spin chain

In this paper, we consider the model proposed by Lakshmanan, Porsezian and Daniel [M. Lakshmanan, K. Porsezian, M. Daniel, Effect of discreteness on the continuum limit of the Heisenberg spin chain, Phys. Lett. A 133 (1988) 483-488] describing the continuum isotropic biquadratic Heisenberg spin chain in one dimensional case. Following the results of integrability in [K. Porsezian, M. Daniel, M. Lakshmanan, On the integrability aspects of the one-dimensional classical continuum isotropic Heisenberg spin chain, J. Math. Phys. 33 (5) (1992) 1807-1816], we prove the existence of periodic weak solutions to this model.     

Discrete skew self-adjoint canonical system and the isotropic Heisenberg magnet model

A discrete analog of a skew self-adjoint canonical (Zakharov-Shabat or AKNS) system with a pseudo-exponential potential is introduced. For the corresponding Weyl function the direct and inverse problem are solved explicitly in terms of three parameter matrices. As an application explicit solutions are obtained for the discrete integrable nonlinear equation corresponding to the isotropic Heisenberg magnet model. State-space techniques from mathematical system theory play an important role in the proofs.     

Anharmonic states of the one-dimensional anisotropic Heisenberg model with free boundary conditions

States described by anharmonic functions are shown to exist in the one-dimensional anisotropic Heisenberg model with a finite number of spins 1/2 and free boundary conditions. Several such states are determined. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 117, No. 3, pp. 489–497, December, 1998.     

Two-magnon states in the one-dimensional isotropic heisenberg model with free boundary conditions

The eigenfunctions and eigenvalues of the energy of two-magnon states in the finite one-dimensional isotropic Heisenberg model S = 1/2 with free boundary conditions were found by solving the Schrödinger equation. The obtained solutions are single-parametric in contrast to two-parametric solutions in the model with cyclic boundary conditions. The amplitudes of the wave functions of coupled two-magnon states exponentially depend on both the distance between the flipped spins and the coordinate of the center of the complex. This leads to a localization of low energy complexes at the ends of the ferromagnetic chain.     

Revisiting the heat kernel on isotropic and nonisotropic Heisenberg groups*

The aim of this paper is threefold. First, we obtain the precise bounds for the heat kernel on isotropic Heisenberg groups by using well-known results in the three-dimensional case. Second, we study the asymptotic estimates at infinity for the heat kernel on nonisotropic Heisenberg groups. As a consequence, we give uniform upper and lower estimates of the heat kernel, and complete its short-time behavior obtained by Beals–Gaveau–Greiner. Third, we prove that the uniform asymptotic behaviour at infinity (so the small-time asymptotic behaviour) of the heat kernel for Grushin operators, obtained by the first author, are still valid in two and three dimensions.     

Three-dimensional vortices in the Heisenberg model

Theoretical and Mathematical Physics - We find an exact solution of the three-dimensional Heisenberg model (an $$SO(3)$$ model in field theory) using a differential substitution. We obtain new...     

Inverse scattering in one-dimensional nonconservative media

The inverse scattering problem arising in wave propagation in one-dimensional non-conservative media is analyzed. This is done in the frequency domain by considering the Schrödinger equation with the potentialikP(x)+Q(x), wherek ² is the energy andP(x) andQ(x) are real integrable functions. Using a pair of uncoupled Marchenko integral equations,P(x) andQ(x) are recovered from an appropriate set of scattering data including bound-state information. Some illustrative examples are provided.Dedicated to M.G. Kreîn, one of the founding fathers of inverse scattering theory.     

The continuum limit of the infinite isotropic Heisenberg chain in its ground state representation

The Hamiltonian of the infinite spin-$\text{1}\text{2}$ isotropic Heisenberg chain with lattice spacing ? and nearest-neighbor interaction, corresponding to a ground state representation of the quasilocal algebra, acts on a Hilbert space that can be naturally embedded in a continuum Fock space. It is proved that as ? → 0 this Hamiltonian converges on the latter space in the strong resolvent sense to a Hamiltonian describing a gas of free non-relativistic equal mass bosons. It is also shown that the projection on the spin wave sectors containing bound magnons strongly converges to zero as ? → 0, even though the bound magnon energies converge as ? → 0.     

Heisenberg model in pseudo-Euclidean spaces

We construct analogues of the classical Heisenberg spin chain model (or the discrete Neumann system), on pseudo-spheres and light-like cones in the pseudo-Euclidean spaces and show their complete Hamiltonian integrability. Further, we prove that the Heisenberg model on a light-like cone leads to a new example of the integrable discrete contact system.