Quasi-exactly solvable Fokker-Planck equations

We consider exact and quasi-exact solvability of the one-dimensional Fokker-Planck equation based on the connection between the Fokker-Planck equation and the Schrödinger equation. A unified consideration of these two types of solvability is given from the viewpoint of prepotential together with Bethe ansatz equations. Quasi-exactly solvable Fokker-Planck equations related to the sl(2)-based systems in Turbiner’s classification are listed. We also present one sl(2)-based example which is not listed in Turbiner’s scheme.     

Quasi-exactly solvable quantum systems with explicitly time-dependent Hamiltonians

For a large class of time-dependent non-Hermitian Hamiltonians expressed in terms linear and bilinear combinations of the generators for an Euclidean Lie-algebra respecting different types of PT-symmetries, we find explicit solutions to the time-dependent Dyson equation. A specific Hermitian model with explicit time-dependence is analyzed further and shown to be quasi-exactly solvable. Technically we constructed the Lewis–Riesenfeld invariants making use of the metric picture, which is an equivalent alternative to the Schrödinger, Heisenberg and interaction picture containing the time-dependence in the metric operator that relates the time-dependent Hermitian Hamiltonian to a static non-Hermitian Hamiltonian.     

New analytically solvable models of relativistic point interactions

Two new analytically solvable models of relativistic point interactions in one dimension (being natural extensions of the nonrelativistic , resp. , interactions) are considered. Their spectral properties in the case of finitely many point interactions as well as in the periodic case are fully analyzed. Moreover, we explicitly determine the spectrum in the case of independent, identically distributed random coupling constants and derive the analog of the Saxon and Hutner conjecture concerning gaps in the energy spectrum of such systems.     

Coulomb effects in relativistic nuclear collisions

We derive simple analytical formulas for Coulomb final-state interactions and apply them to the analysis of recent data on nuclear collisions. The π^?/π⁺ ratio, the π⁺ inclusive cross section, and the n/p ratio are studied. A relativistic field theoretic model is used to derive the formulas to first order in Zα. Using well-known non-perturbative results, we recast those formulas into an approximate non-perturbative form valid when finite-size effects are negligible. This allows us to calculate the important k → 0 limit. The final formulas are covariant and take into account multiple independently moving charged fragments of finite size and finite thermal expansion velocities. Our studies demonstrate analytically the complexity and importance of Coulomb distortions in nuclear collisions.     

One-electron relativistic molecules with Coulomb interaction

As an approximation to a relativistic one-electron molecule, we study the operator \(H = ( - \Delta + m^2 )^{1/2} - e^2 \sum\limits_{j = 1}^K {Z_j } |x - R_j |^{ - 1}\) withZ _j ≧0,e ^?2=137.04.H is bounded below if and only ife ² Z _j ≦2/π allj. Assuming this condition, the system is unstable whene ²∑Z _j >2/π in the sense thatE ₀=inf spec(H)→?∞ as the R _j →0, allj. We prove that the nuclear Coulomb repulsion more than restores stability; namely \(E_0 + 0.069e^2 \sum\limits_{i< j} {Z_i Z_j } |R_i - R_j |^{ - 1} \geqq 0\) . We also show thatE ₀ is an increasing function of the internuclear distances |R _i ?R _j |.     

Quasi-exactly solvable decatic model description of nuclei near the X(5) critical point

The Bohr Hamiltonian with axially deformed shape confined in a quasi-exactly solvable decatic β-part potential is studied.It is shown that the decatic model can well reproduce the X(5)model results as far as the energy ratios in the ground and beta band and related B(E2)values are concerned.Fitting results to the low-lying energy ratios and relevant B(E2)values of even-even X(5)candidates ¹⁵⁰Nd,¹⁵⁶Dy,¹⁶⁴Yb,¹⁶⁸Hf,¹⁷⁴Yb,^176,178,180Os,and ^188,190Os show that the decatic model provides the best fitting results for the energy ratios,while the X(5)model is the best at reproducing the B(E2)values of these nuclei,in which the beta-bandhead energy is lower than that of the gamma band.While for even-even nuclei,such as ^154,156,158Gd,with bandhead energies of the beta and gamma bands more or less equal within the X(5)critical point to the axially deformed region,our numerical analysis indicates that the decatic model is better than the X(5)model in describing both the low-lying level energies and related B(E2)values.     

Exactly solvable one-dimensional inhomogeneous models

We present a simple way of constructing one-dimensional inhomogeneous models (random or quasiperiodic) which can be solved exactly. We treat the example of an Ising chain in a varying magnetic field, but our procedure can easily be extended to other one-dimensional inhomogeneous models. For all the models we can construct, the free energy and its derivatives with respect to temperature can be computed exactly at one particular temperature.     

The generalized Coulomb interactions for relativistic scalar bosons

Approximate analytical solutions of Duffin–Kemmer–Petiau (DKP) equation are obtained for the truncated Coulomb, generalized Cornell, Richardson and Song–Lin potentials via the quasi-exact analytical ansatz approach.     

Class of exactly solvable pairing models

We present three classes of exactly solvable models for fermion and boson systems, based on the pairing interaction. These models are solvable in any dimension. As an example we show the first results for fermions interacting with repulsive pairing forces in a two-dimensional square lattice. In spite of the repulsive pairing force the exact results show attractive pair correlations.     

Exactly solvable models showing chaotic behavior

Several examples of the one-dimensional mapping which are exactly solvable and show chaotic behaviour are presented. The importance of the accuracy of the numerical calculation is stressed.     

The nuclear Coulomb sum rule in a relativistic model

The nuclear Coulomb sum rule is investigated in a relativistic quantum field theory of the nucleus based on baryons and mesons. First an effective, local, covariant, conserved electromagnetic current operator is constructed for the many-baryon system. It describes the electromagnetic structure of an isolated nucleon; the lowest-mass two-pion contribution to the spectral weight functions of the form factors is contained in it. The sum rule is then evaluated in a model based on baryons and neutral scalar and vector mesons. In the mean-field approximation (MFT) this model correctly describes the saturation properties of nuclear matter. The “one-body” term in the sum rule can be evaluated exactly through the use of the canonical anticommutation relations for the baryon field and the identification of conserved quantities. The remaining relativistic two-body contribution is evaluated in the MFT. Meson contributions to the sum rule at large momentum transfers $\text{q}\text{2k}{}_{\mathrm{<mtext>F</mtext>}}\text{? 1}$ completely dominate anticipated static, short-range, two-nucleon correlation contributions to the non-relativistic Coulomb sum rule. One possible implication is that the nucleus must (at least) be considered as a dynamic system of mesons and baryons.     

Supermultiplicity and the relativistic Coulomb problem with arbitrary spin

The Hamiltonian for n relativistic electrons without interaction but in a Coulomb potential is well known. If in this Hamiltonian we take r _′ ^u =r′, P _′ ^u =P′ with u=1,2,..., n, we obtain a one-body problem in a Coulomb field, but the appearance of n of the α _u , u=1,..., n, each of which corresponds to spin $\tfrac{1}{2}$ , indicates that we may have spins up to (n/2). We analyze this last problem first by denoting the 4×4 matrices α, β as direct products of 2×2 matrices which correspond to the ordinary spin, and a new concept, also related to the SU(2) group, which we call sign spin. In this new notation our problem depends on the sixteen generators of a U(4) group reduced along the chain Û(2)??(2) sub-groups associated with the ordinary and sign spins. We now make a change of variables in our Hamiltonian so a term ε related to the frequency ω of an oscillator, which will be our variational parameter, appears in it, and later construct the full states of the problem with a harmonic oscillator of frequency 1 and ordinary and sign spin parts. Finally we obtain the matrix representation of our Hamiltonian with respect to the states mentioned and discuss the energy spectra of the problem where the partition {h} representing the irrep of U(4) and j the total angular momentum, take the values {h}=[1], j= $\tfrac{1}{2}$ ; {h}=[11], j=0; {h}=[2], j=0.     

Exactly solvable models for atom-molecule Hamiltonians

We present a family of exactly solvable generalizations of the Jaynes-Cummings model involving the interaction of an ensemble of SU(2) or SU(1,1) quasispins with a single boson field. They are obtained from the trigonometric Richardson-Gaudin models by replacing one of the SU(2) or SU(1,1) degrees of freedom by an ideal boson. The application to a system of bosonic atoms and molecules is reported.     

Exactly solvable two-dimensional quantum spin models

A method is proposed for constructing an exact ground-state wave function of a two-dimensional model with spin 1/2. The basis of the method is to represent the wave function by a product of fourth-rank spinors associated with the nodes of a lattice and the metric spinors corresponding to bonds between nearest neighbor nodes. The function so constructed is an exact wave function of a 14-parameter model. The special case of this model depending on one parameter is analyzed in detail. The ground state is always a nondegenerate singlet, and the spin correlation functions decay exponentially with distance. The method can be generalized for models with spin 1/2 to other types of lattices. Zh. éksp. Teor. Fiz. 115, 249–267 (January 1999)     

Lattice gases and exactly solvable models

We detail the construction of a family of lattice gas automata based on a model of 't Hooft, proceeding by use of symmetry principles to define first the kinematics of the model and then the dynamics. A spurious conserved quantity appears; we use it to effect a radical transformation of the model into one whose spacetime configurations are equivalent to the two-dimensional states of an exactly solvable statistical mechanics model, the symmetric eight-vertex model with parameters restricted to a disorder variety. We comment on the implications of this identification for the original lattice gas.     

Exactly solvable models of adaptive networks

A satisfiability-unsatisfiability (SAT-UNSAT) transition takes place for many optimization problems when the number of constraints, graphically represented by links between variables nodes, is brought above some threshold. If the network of constraints is allowed to adapt by redistributing its links, the SAT-UNSAT transition may be delayed and preceded by an intermediate phase where the structure self-organizes to satisfy the constraints. We present an analytic approach, based on the recently introduced cavity method for large deviations, which exactly describes the two phase transitions delimiting this adaptive intermediate phase. We give explicit results for random bond models subject to the connectivity or rigidity percolation transitions, and compare them with numerical simulations.     

First-order relativistic corrections to the theory of Coulomb excitation

It is suggested that relativistic effects may be significant in high-precision Coulomb-excitation experiments. First-order corrections to the non-relativistic theory of Coulomb excitation are determined with the restriction that all of the states involved in the excitation have the same parity and that the scattering angle is 180°. The results of some numerical calculations are presented to give an indication of the magnitude of the correction. A recent experiment to determine the B(E2;0⁺ → 2⁺) and excited-state quadrupole moment of ¹⁹⁸Hg is reanalyzed and the relativistic correction to the B(E2) is found to be several times the experimental uncertainty. Sizeable effects are also found in multiple Coulomb excitation.