An exactly solvable anharmonic Bohr hamiltonian and its equivalent boson hamiltonian

The eigenenergies and eigenfunctions of the bound states of an anharmonic quadrupole oscillator are derived in closed form. An interacting boson model hamiltonian is found which has the same eigenspectrum.     

A class of exactly solvable Fokker Planck equations

A class of nonlinear,n-dimensional Fokker Planck equations with exact time dependent solution is presented. An equation of this class can be obtained from any function (q ₁, ,q _n ). Some examples are discussed. For a certain subclass, the associated Itô and Stratonovich stochastic differential equations coincide.     

An exactly solvable impurity model

We present an integrable model describing the interaction of conduction electrons with an arbitrary number of spin-1/2 impurities in one-dimension. The solution of the model is obtained by means of the Bethe function in the case when the number of particles is not conserved. The impurity magnetization, the Kondo-limit, the specific-heat coefficient are calculated on the basis of the Bethe ansatz solution.     

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.     

New type of exactly solvable potential

It is shown that, starting from an exactly solvable potential and making use of the theory of a system of coupled differential equations, it is possible to construct a new type of second-generation potential which is also exactly solvable.     

Classification of exactly solvable potential problems

A differential equation with a known solution is transformed by changing both its dependent and independent variables, and the resulting nonlinear differential equation is then compared with the Schrödinger equation. The method is demonstrated using the confluent hypergeometric differential equation and the solutions to hydrogen, SHO and l=0 Morse potential problems are obtained.     

Equidistant spectra of anharmonic oscillators

Some representative potentials of the anharmonic-oscillator type are constructed. Some corresponding spectra-shift operators are also constructed. These operators are a natural generalization of Fok creation and annihilation operators. The Schrodinger problem for these potentials leads to an equidistant energy spectrum for all excited states, which are separated from the ground state by an energy gap. The general properties of the dynamic system generated by spectral-shift operators of third degree are analyzed. Several examples of such anharmonic oscillators are discussed. The relationship between the eigenvectors of the Schrodinger problem and a certain type of nonclassical orthogonal polynomials is established.     

An exactly solvable non-linear Boltzmann equation

The initial value problem for a model Boltzmann equation of a two dimensional gas with a continuous or discrete energy distribution function and a transition probability δ(? - ?') is solved exactly; ? and ?' are the total energies before and after collision.     

An exactly solvable two-dimensional biomembrane model

A model for a two-dimensional lipid bilayer, in which both short-range repulsive forces and long-range attractive forces piay a role, is solved exactly. First, in the absence of attractive forces, the configurational entropy is calculated by restricting the lipid hydrocarbon tails to a lattice and solving the resulting counting problem exactly. When long-range attractive forces which decrease exponentially with distance are acting between the polar head groups of the lipids, the partition function can still be calculated exactly, using integration over Gaussian random fields. The thermodynamic functions show a singularity which reflects the completion of the process of self-assemblage of the lipid bilayer. Finally, the model is used to test an approximation method for lipid bilayers.     

A quantum exactly solvable non-linear oscillator with quasi-harmonic behaviour

The quantum version of a non-linear oscillator, previously analyzed at the classical level, is studied. This is a problem of quantization of a system with position-dependent mass of the form m = (1 + λx²)⁻¹ and with a λ-dependent non-polynomial rational potential. This λ-dependent system can be considered as a deformation of the harmonic oscillator in the sense that for λ → 0 all the characteristics of the linear oscillator are recovered. First, the λ-dependent Schrödinger equation is exactly solved as a Sturm-Liouville problem, and the λ-dependent eigenenergies and eigenfunctions are obtained for both λ > 0 and λ < 0. The λ-dependent wave functions appear as related with a family of orthogonal polynomials that can be considered as λ-deformations of the standard Hermite polynomials. In the second part, the λ-dependent Schrödinger equation is solved by using the Schrödinger factorization method, the theory of intertwined Hamiltonians, and the property of shape invariance as an approach. Finally, the new family of orthogonal polynomials is studied. We prove the existence of a λ-dependent Rodrigues formula, a generating function and λ-dependent recursion relations between polynomials of different orders.     

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.     

A set of exactly solvable Ising models with half-odd-integer spin

We present a set of exactly solvable Ising models, with half-odd-integer spin-S on a square-type lattice including a quartic interaction term in the Hamiltonian. The particular properties of the mixed lattice, associated with mixed half-odd-integer spin-(S,1/2) and only nearest-neighbor interaction, allow us to map this system either onto a purely spin-1/2 lattice or onto a purely spin-S lattice. By imposing the condition that the mixed half-odd-integer spin-(S,1/2) lattice must have an exact solution, we found a set of exact solutions that satisfy the free fermion condition of the eight vertex model. The number of solutions for a general half-odd-integer spin-S is given by S+1/2. Therefore we conclude that this transformation is equivalent to a simple spin transformation which is independent of the coordination number.     

Interpretation of an exactly solvable two-component plasma

The interpretation of the exact calculation of the partition function and correlations of a two-component plasma obtained earlier is considered. The system has species of charge ratio 12 which are constrained to lie on a circle and interact via the two-dimensional Coulomb potential. By studying the exact results we gain an understanding of why the excess thermodynamic quantities of the two component system can be well approximated by the sum of the appropriate excess thermodynamic quantities of the one-component systems.     

Entanglement and relaxation in exactly solvable models

A system put in contact with a large heat bath normally thermalizes. This means that the state of the system ρ^ℐ(t) approaches an equilibrium state ρ_eq^ℐ, the latter depending only on macroscopic characteristics of the bath (e.g. temperature), but not on the initial state of the system. The above statement is the cornerstone of the equilibrium statistical mechanics; its validity and its domain of applicability are central questions in the studies of the foundations of statistical mechanics. In the present contribution we discuss the recently proven general theorems about thermalization and demonstrate how they work in exactly solvable models. In particular, we review a necessary condition for the system initial state independence (ISI) of ρ_eq^ℐ, which was proven in our previous work, and apply it for two exactly solvable models, the XX spin chain and a central spin model with a special interaction with the environment. In the latter case we are able to prove the absence of the system ISI. Also the Eigenstate Thermalization Hypothesis is discussed. It is pointed out that although it is supposed to be generically true in essentially not-integrable (chaotic) quantum systems, it is how-ever also valid in the integrable XX model.     

Perturbation theory of odd anharmonic oscillators

We study the perturbation theory forH=p ²+x ²+x ²ⁿ⁺¹,n=1, 2, .... It is proved that when Im0,H has discrete spectrum. Any eigenvalue is uniquely determined by the (divergent) Rayleigh-Schrödinger perturbation expansion, and admits an analytic continuation to Im=0 where it can be interpreted as a resonance of the problem.Partially supported by G.N.F.M., C.N.R.     

A class of exact solutions for doubly anharmonic oscillators

The solution of a difference equation in the form of an infinite continued fraction is used to obtain a class of exact solutions for the eigenfunctions and eigenvalues of doubly anharmonic oscillators described by potentials of the type (1/2)²x²+(1/4)x⁴+(1/6)x⁶, n>0, provided certain constraints on the couplings are satisfied. The class is denumerably infinite but not complete.