Quasi-exact solvability beyond the <Emphasis Type="Italic">sl</Emphasis>(2) algebraization

We present evidence to suggest that the study of one-dimensional quasi-exactly solvable (QES) models in quantum mechanics should be extended beyond the usual sl(2) approach. The motivation is twofold: We first show that certain quasi-exactly solvable potentials constructed with the sl(2) Liealgebraic method allow for a new larger portion of the spectrum to be obtained algebraically. This is done via another algebraization in which the algebraic Hamiltonian cannot be expressed as a polynomial in the generators of sl(2). We then show an example of a new quasi-exactly solvable potential which cannot be obtained within the Lie algebraic approach. The text was submitted by the authors in English.     

On shape variation of confined triatomics of XY<Subscript>2</Subscript>-type

Harmonic confinement of initially isolated symmetric triatomic molecules can induce a transition from a bent, directed bond-type structure to helium-like angular correlation of the two equal particles. In an exactly solvable modification of the Hooke-Calogero model it is demonstrated that there is a well-defined transition between the two cases if the confinement strength is increased. Furthermore confinement is shown to reduce the system’s effective diameter, which at the transition point has shrunk by 26% in comparison to the isolated system.     

Complex oscillator and Painlevé IV equation

Supersymmetric quantum mechanics is a powerful tool for generating exactly solvable potentials departing from a given initial one. In this article the first- and second-order supersymmetric transformations will be used to obtain new exactly solvable potentials departing from the complex oscillator. The corresponding Hamiltonians turn out to be ruled by polynomial Heisenberg algebras. By applying a mechanism to reduce to second the order of these algebras, the connection with the Painlevé IV equation is achieved, thus giving place to new solutions for the Painlevé IV equation.     

The conditionally solvable natanzon potentials

It is shown how to derive the recently introduced conditionally solvable Natanzon potentials in the framework of the Schrödinger’s formulation of quantum mechanics. Their relation to the exactly solvable potentials is considered. It is shown that in this way the variety of conditionally solvable potentials can be obtained. The relationship between the Natanzon potentials and shape invariant ones is derived.     

The (0,2) exactly solvable structure of chiral rings,Landau-Ginzburg theories and Calabi-Yau manifolds

We identify the exactly solvable theory of the conformal fixed point of (0,2) Calabi-Yau σ-models and their Landau-Ginzburg phases. To this end we consider a number of (0,2) models constructed from a particular (2,2) exactly solvable theory via the method of simple currents. In order to establish the relation between exactly solvable (0,2) vacua of the heterotic string, (0,2) Landau-Ginzburg orbifolds and (0,2) Calabi-Yau manifolds, we compute the Yukawa couplings of the exactly solvable model and compare the results with the product structure of the chiral ring which we extract from the structure of the massless spectrum of the exact theory. We find complete agreement between the two up to a finite number of renormalizations. For a particularly simple example we furthermore derive the generating ideal of the chiral ring from a (0,2) linear σ-model which has both a Landau-Ginzburg and a (0,2) Calabi-Yau phase.     

Random matrix theories and exactly solvable models

A connection between random-matrix theories and exactly solvable models is discussed here. This is done in three parts: firstly, for theWigner—Dyson case; secondly, for the short-range Dyson case; and thirdly, for the pseudo-Hermitian one. The exactly solvable models are variants and extensions of Calogero—Sutherland—Moser systems.     

Step Operators for the Quasi-exactly Solvable Systems

In this paper, we investigate the step operators for the quasi-exactly solvable problems. We also discuss the commutation relations between the step operators and the Hamiltonian of the quasi-exactly solvable system. After obtaining the general results, we take the anharmonic oscillators with x ⁶ anharmonicity in quasi-exactly solvable problems as examples to give the specific forms of step operators.     

Solutions of nonrelativistic Schrödinger equation from relativistic Klein-Gordon equation

The relativistic one-dimensional Klein-Gordon equation can be exactly solved for a certain class of potentials. But the nonrelativistic Schrödinger equation is not necessarily solvable for the same potentials. It may be possible to obtain approximate solutions for the inexact nonrelativistic potential from the relativistic exact solutions by systematically removing relativistic portion. We search for the possibility with the harmonic oscillator potential and the Coulomb potential, both of which can be exactly solvable nonrelativistically and relativistically. Though a rigorous algebraic approach is not deduced yet, it is found that the relativistic exact solutions can be a good starting point for obtaining the nonrelativistic solutions.     

Thermodynamics’ third law and quantum phase transitions

We report on the fact that microscopically enforcing fulfillment of thermodynamics’ third law on a system of fermions automatically yields the values of the external parameter (here coupling strengths in the pertinent Hamiltonian) at which quantum phase transitions take place. Our considerations are illustrated via an exactly solvable model of Plastino and Moszkowski [Il Nuovo Cimento 47, 470 (1978)].     

Exactly solvable two-dimensional stationary Schrödinger operators obtained by the nonlocal Darboux transformation

The Fokker–Planck equation associated with the two-dimensional stationary Schrödinger equation has the conservation law form that yields a pair of potential equations. The special form of Darboux transformation of the potential equations system is considered. As the potential variable is a nonlocal variable for the Schrödinger equation that provides the nonlocal Darboux transformation for the Schrödinger equation. This nonlocal transformation is applied for obtaining of the exactly solvable two-dimensional stationary Schrödinger equations. The examples of exactly solvable two-dimensional stationary Schrödinger operators with smooth potentials decaying at infinity are obtained.     

Countably Solvable Connected Pro-Lie Groups Are <Emphasis Type="Italic">u</Emphasis>-Amenable

In contrast to some well-known discrete groups, countably solvable connected pro-Lie groups are u-amenable in the sense of de la Harpe’s 1973 paper.     

Painleve Integrability,Consistent Riccati Expansion Solvability and Interaction Solution for the Coupled mKdV-BLMP System

The integrability of the coupled, modified KdV equation and the potential Boiti-Leon-Manna-Pempinelli(mKdVBLMP) system is investigated using the Painleve analysis approach. It is shown that this coupled system possesses the Painleve property in both the principal and secondary branches. Then, the consistent Riccati expansion(CRE)method is applied to the coupled mKdV-BLMP system. As a result, it is CRE solvable for the principal branch while non-CRE solvable for the secondary branch. Finally; starting from the last consistent differential equation in the CRE solvable case, soliton, multiple resonant soliton solutions and soliton-cnoidal wave interaction solutions are constructed explicitly.     

Mass Generation in a Fermionic Model with Finite Range Time Dependent Interactions

Bardeen, Cooper and Schrieffer in their paper on the theory of superconductivity introduced a model of interacting fermions (BCS model) in which the (instantaneous) interaction is only between electrons of opposite momentum and spin (Cooper pairs). Subsequently it was claimed that in the thermodynamic limit the BCS model is equivalent to the (exactly solvable) quadratic mean field BCS model in which the phenomenon of mass generation is present; a rigorous proof of this equivalence is however still an open problem. In this paper we consider an interacting fermionic model in which the Cooper pairs interact through a finite range time dependent interaction. For this model (quartic in the fermions and not solvable) we are able to prove the generation of mass in the thermodynamic limit and its equivalence with the mean field BCS model. The proof is achieved by a convergent perturbation expansion about mean field theory.     

Zur Theorie der stationären Bewegung von Ionen im elektrischen Feld

In a static electric field a one-dimensional model of the motion of ions, controlled solely by charge exchange (electron capture from gas moleculs) is considered. Solutions of the kinetic equastions for ions and fast neutrals are derived in several special cases. If the charge exchange cross section is constant, the kinetic equation is easily solvable in a closed form; if the cross section is a function of ion energy, however, the problem is reducible to a simple integral equation for the energy distribution function.     

symmetries of the Zn × Zn Belavin Model

It is shown that there exist some symmetries in the Z_n × Z_n Belavin model. These symmetric properties can be used to construct the new exactly solvable statistical model with nontrivial boundary terms.     

Envelope self-similar solutions for the nonautonomous and inhomogeneous nonlinear Schrödinger equation

By means of the similarity transformation connecting with the solvable stationary equation, the self-similar combined Jacobian elliptic function solutions and fractional form solutions of the generalized nonlinear Schrödinger equation (NLSE) are obtained when the dispersion, nonlinearity, and gain or absorption are varied. The propagation dynamics in a periodic distributed amplification system is investigated. Self-similar cnoidal waves and corresponding localized waves including bright and dark similaritons (or solitons) for NLSE and arch and kink similaritons (or solitons) for cubic-quintic NLSE are analyzed. The results show that the intensity and the width of chirped cnoidal waves (or similaritons) change more distinctly than that of chirp-free counterparts (or solitons).     

A four-level “Ξ-system”: Exact solvability and effective evolution operators via multiple scales

Properties of a four-level atomic system interacting with one and two modes of the electromagneticfield in a “Ξ”-configuration are investigated. By linearization of the Hamiltonians we show that the corresponding mathematical models are exactly solvable. To obtain simpler effective Hamiltonians the perturbative method of multiple scales is applied. The lowest-order corrections to the resulting effective evolution operators are also calculated.     

Application of quasiexactly solvable potential method to the <Emphasis Type="Italic">N</Emphasis>-body problem of anharmonic oscillators

The quasiexactly solvable potential method is used to determine the energies and the corresponding exact eigenfunctions for a system of N particles with equal mass interacting via an anharmonic potential. For systems with five and seven particles, we compute the ground state and the first excited state only, and compare the spectrums with the results obtained by Ritz approximation method.     

Pressures for a One-Component Plasma on a Pseudosphere

The classical (i.e., non-quantum) equilibrium statistical mechanics of a two-dimensional one-component plasma (a system of charged point-particles embedded in a neutralizing background) living on a pseudosphere (an infinite surface of constant negative curvature) is considered. In the case of a flat space, it is known that, for a one-component plasma, there are several reasonable definitions of the pressure, and that some of them are not equivalent to each other. In the present paper, this problem is revisited in the case of a pseudosphere. General relations between the different pressures are given. At one special temperature, the model is exactly solvable in the grand canonical ensemble. The grand potential and the one-body density are calculated in a disk, and the thermodynamic limit is investigated. The general relations between the different pressures are checked on the solvable model.     

Exactly solvable associated Lamé potentials and supersymmetric transformations

A systematic procedure to derive exact solutions of the associated Lamé equation for an arbitrary value of the energy is presented. Supersymmetric transformations in which the seed solutions have factorization energies inside the gaps are used to generate new exactly solvable potentials; some of them exhibit an interesting property of periodicity defects.