Solution of the problem of indirect spin-spin interaction via the phonon field

A new transformation which differs from the famous one of Fröhlich is introduced to transform the hamiltonian of the interacting spin system and phonon system to a representation which leads to the exact expression for the operator H_SS of the indirect spin-spin interaction via the phonon field. The evident form of the operator H_SS is determined.     

Extending integrable hamiltonian systems from 2 to N dimensions

We describe a method of constructing N-dimensional integrable hamiltonian systems starting from two-dimensional ones. Several models are examined. Included are the two candidates for integrability discovered by Lakshmanan and Sahadevan for which we find the integrals of motion. Results for other N-dimensional integrable hamiltonian systems are also presented.     

Lax form of the quantum mechanical eigenvalue problem

The quantum mechanical eigenvalue problem with the hamiltonian H = H₀ + tV is written as a set of dynamical equations for the eigenvalues x_n(t) and the matrix elements V_nm(t) regarding the parameter t as time. By appropriate changes of variables it can be expressed as a pair of matrix equations with the Lax form, hence we are able to write all the possible constants of the motion explicitly. Implications of these constants to the statistical properties of levels are discussed.     

Pauli fermions as components of D=2 supersymmetrical quantum mechanics

We consider the Pauli hamiltonian for charged fermions in electromagnetic and scalar potentials as being a component of the D=2 supersymmetrical hamiltonian. The admissible class of electromagnetic potentials is described. For a subclass of such potentials the fermion spectrum consists of pairs of equally splitted levels. The double degeneracy of levels (the sign of hidden supersymmetry) arises for holomorphic superpotentials.     

Quantum field theory on Riemann surfaces and the unitarity problem

By the use of the Klein method instead of the theta-function method of Jacobi we are able to relate a conformal quantum theory or Riemann surfaces to the corresponding flat-space field theory and its Virasoro algebra. Physical positivity holds on a distinguished real subset in the manifold with nonrivial Hausdorff dimension which in the general case g > 1 cannot be shifted by a hamiltonian. Our picture of obtaining curved two-dimensional quantum field theories by applying a special diffeomorphism to flat ones resembles that of the Hawking-Unruh effect.     

N = 2 superconformal algebra and integrable O(2) fermionic extensions of the Korteweg-de Vries equation

The N = 2 superconformal algebra is shown to be related to the second hamiltonian structure of three integrable fermionic extensions of the Korteweg-de Vries equation. One of these systems is bi-hamiltonian but not supersymmetric while the reverse is true for the other two.     

On the use of ibm-2 in a shell of neutrons and protons

We discuss the role of isospin in boson approximations for a system with neutrons and protons in the same shell. The lack of isospin symmetry in the IBM-2 model can be approximately corrected by using an N, T-dependent hamiltonian.     

IBM-2 description of M1 properties in deformed nuclei

Results of schematic calculations are presented in which various terms breaking F-spin symmetry are considered in the hamiltonian of the neutron-proton interacting boson model (IBM-2). Specific attention is paid to the effect of F-spin symmetry breaking on γ → ground and γ → γ M1 transitions in deformed nuclei. A comparison with available M1 data in the rare-earth nuclei is presented. The constraints implied by these data on the form of the IBM-2 hamiltonian in well-deformed nuclei are discussed.     

Polarization dynamics for spin 12-spin I systems with Breit-Rabi interaction

We study the full polarization dynamics of two interacting systems of spin $\text{1}\text{2}$ and spin I for the Breit-Rabi hamiltonian, the sum of contact hyperfine and Zeeman terms. We obtain the time dependence of the polarizations by solving the eigenvalue problems for the interaction hamiltonian and discuss some applications.     

Time-dependent variational principle on the group SO(2r): Generalizations of time-dependent Hartree-Fock

The family of all Hartree-Fock-Bogoliubov (HFB) states for a given set of r isosopin-spin orbitals form a set of coherent states. The set of antisymmetrized geminal power (AGP) states for a given set of r isospin-spin orbitals form a set of charge-projected coherent states, with the number of particles n as the “charge” and the HFB coherent state as the generating function. Both these coherent states are associated with the group SO(2r). The approximate time evolution of the system generated by restricting the quantum mechanical evolution to the family of HFB and AGP coherent states is described as a classical dynamics with the energy of the coherent state as hamiltonian function. The phase space is isomorphic to the coset space SO(2r)/U(r). The random phase approximation based on HFB and AGP states is derived by considering the harmonic approximation to the hamiltonian function. This work generalizes the group theoretical approaches to Hartree-Fock, and time-dependent Hartree-Fock theory by the use of non-number-conserving (HFB) and correlated (AGP) states.     

The exact solution of 2D ZN invariant statistical models

The exact solution of the hamiltonian version of Z_N invariant statistical models at τ ≠ 0 is found by means of the Bethe-ansatz technique.     

Hamiltonian dynamics of a charged fluid,including electro- and magnetohydrodynamics

Nondissipative hydrodynamic equations for a charged, polarized and magnetized nonrelativistic fluid moving in a self-consistent electromagnetic field are presented and shown to possess a hamiltonian structure, associated to the dual of a certain Lie algebra of semidirect-product type. Ideal magnetohydrodynamics and electrohydrodynamics both emerge in hamiltonian form as regular limits (i.e., special cases) of the hamiltonian structure for the more general theory.     

Gauge covariant formulation of the generating operator. 2. Systems on homogeneous spaces

A gauge covariant approach to the operator Λ, generating the n-wave type equations on homogeneous spaces is proposed. The operator Λ̃ for the gauge equivalent equations is explicitly constructed. The main results (such as conservation laws, hierarchies of hamiltonian structures, etc.) for the n-wave type equations and their gauge equivalent ones are formulated in terms of Λ and Λ̃ respectively.     

Studies of the ground-state properties of the Lipkin-Meshkov-Glick model via the atomic coherent states

The ground-state energy of the Lipkin-Meshkov-Glick hamiltonian was estimated by the Bogoliubov-Lieb inequalities which involve the atomic coherent states. It was found that the so-called Q-representation of the pseudospin hamiltonian provides a good estimation of the ground-state energy. Furthermore, nuclear phase transitions can be vividly described by the deviations of the pseudospin from the southpole of the so-called Bloch sphere.     

A new theoretical approach to study the effects of active molecules on lipid bilayer properties: The cholesterol problem

A model to study phospholipid bilayers with active molecules that greatly disturb molecular conformations of their neighboring lipids is proposed. In the particular case of cholesterol in DPPC bilayers, this method allows us to use the spin 1 Ising hamiltonian as a natural extension of the two state model of chain melting transition. Physical processes responsible for the decreasing of the total transition enthalpy Δ_tQ and temperature T_c, and for the phase separation, both below and above T_c, when cholesterol is added, are easily identified. From the details of the phase separation, the two peak specific heat data are explained. Similar models for the addition of other binding molecules to lipid bilayers are discussed.     

On the Hartree-Fock time-dependent problem

A previous result is generalized. An existence and uniqueness theorem is proved for the Hartree-Fock time-dependent problem in the case of a finite Fermi system interacting via a two body potential, which is supposed dominated by the kinetic energy part of the one-particle hamiltonian.     

Mixing coefficients of the lattice weak hamiltonian with dimension five operators

We compute the mixing coefficients of the four fermion operators which appear in the non-leptonic ΔS=1, octet hamiltonian with operators of dimension five. The calculation completes the construction of the perturbative part of the weak hamiltonian, to be used in Monte Carlo simulations of lattice QCD with Wilson fermions.     

Statistical approach to the antiferromagnetic ground state problem

The statistical approach is adopted in an attempt to find the antiferromagnetic ground state. As a result an eigenstate of the Heisenberg hamiltonian is found which is not the true ground state, but which could serve as a useful starting point for other methods.The author is grateful to ing. P. Novák for valuable discussions.