On the equivalence of the Schrödinger and the quantum Liouville equations

We present a semigroup analysis of the quantum Liouville equation, which models the temporal evolution of the (quasi) distribution of an electron ensemble under the action of a scalar potential. By employing the density matrix formulation of quantum physics we prove that the quantum Liouville operator generates a unitary group on L² if the corresponding Hamiltonian is essentially self-adjoint. Also, we analyse the existence and non-negativity of the particale density and prove that the solutions of the quantum Liouville equation converge to weak solutions of the classical Liouville equation as the Planck constant tends to zero (assuming that the potential is sufficiently smooth).     

On the expected discounted penalty function at ruin of a surplus process with interest

In this paper, we study the expected value of a discounted penalty function at ruin of the classical surplus process modified by the inclusion of interest on the surplus. The ‘penalty’ is simply a function of the surplus immediately prior to ruin and the deficit at ruin. An integral equation for the expected value is derived, while the exact solution is given when the initial surplus is zero. Dickson’s [Insurance: Mathematics and Economics 11 (1992) 191] formulae for the distribution of the surplus immediately prior to ruin in the classical surplus process are generalised to our modified surplus process.     

Asymptotic analysis of the quantum Liouville equation

We present an asymptotic analysis of the quantum Liouville equation with respect to the Planck's constant, which models the temporal evolution of the (quasi)distribution of an ensemble of electrons under the action of a potential. We consider two cases: firstly a smooth potential, and secondly a potential modelled by a δ-distribution. In both cases the zeroth-order term behaves classically. In the smooth case the classical Liouville equation is satisfied and in the case for the δ-potential an interface condition is derived, so that everything is reflected at the potential barrier.     

Equivalence of Many-Gluon Green’s Functions in the Duffin-Kemmer-Petieu and Klein-Gordon-Fock Statistical Quantum Field Theories

We prove the equivalence of many-gluon Green’s functions in the Duffin-Kemmer-Petieu and Klein-Gordon-Fock statistical quantum field theories. The proof is based on the functional integral formulation for the statistical generating functional in a finite-temperature quantum field theory. As an illustration, we calculate one-loop polarization operators in both theories and show that their expressions indeed coincide.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 3, pp. 368–374, June, 2005.     

Third-order linear differential equation with three additional conditions and formula for Green’s function

In this paper, we investigate a third-order linear differential equation with three additional conditions. We find a solution to this problem and give a formula and an existence condition for Green’s function. We compare two Green’s functions for two such problems with different additional conditions: nonlocal and classical boundary conditions. Formula applications are shown by examples.     

S-matrix description of nonequilibrium finite-temperature systems

We consider the “inclusive” (“partial”) method for describing nonequilibrium dissipative systems at the early (kinetic) evolution stage, when the temperature distribution is nonuniform. We formulate the perturbation theory in terms of space-time-local temperature Green’s functions and derive the Liouville equation for the one-particle partition function. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 3, pp. 368–380, December, 2006.     

Runge–Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments

In this paper we deal with the numerical solutions of Runge–Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments. The numerical solution is given by the numerical Green’s function. It is shown that Runge–Kutta methods preserve their original order for first-order periodic boundary value differential equations with piecewise constant arguments. We give the conditions under which the numerical solutions preserve some properties of the analytic solutions, e.g., uniqueness and comparison theorems. Finally, some experiments are given to illustrate our results.     

Quantum group structure and local fields in the algebraic approach to 2D gravity

This review contains a summary of the work by J.-L. Gervais and the author on the operator approach to 2d gravity. Special emphasis is placed on the construction of local observables — the Liouville exponentials and the Liouville field itself — and the underlying algebra of chiral vertex operators. The double quantum group structure arising from the presence of two screening charges is discussed and the generalized algebra and field operators are derived. In the last part, we show that our construction gives rise to a natural definition of a quantum tau function, which is a noncommutative version of the classical group-theoretic representation of the Liouville fields by Leznov and Saveliev.Supported by DFG.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 1, pp. 158–191, July, 1995.     

New exact solutions to the nonautonomous Liouville equation

We obtain new formulas for the exact analytic solutions to the nonautonomous elliptic Liouville equation in the two-dimensional coordinate space with the free function dependent specially on an arbitrary harmonic function. We present new exact solutions to the wave Liouville equation with two arbitrary functions, providing original formulas for the general solution for the classical (autonomous) and wave Liouville equations. Some equivalence transformations are presented for the elliptic Liouville equation depending on conjugate harmonic functions. In particular, we indicate a transformation that reduces the equation under study to an autonomous form.     

Green’s functions for stationary problems with nonlocal boundary conditions

In this paper, we investigate Green’s functions for various stationary problems with nonlocal boundary conditions. We express the Green’s function per Green’s function for a problem with classical boundary conditions. This property is illustrated by various examples. Properties of Green’s functions with nonlocal boundary conditions are compared with those for classical problems. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-73/09.     

Some Spectral Properties of One Sturm-Liouville Type Problem with Discontinuous Weight

We consider a discontinuous weight Sturm-Liouville equation together with eigenparameter dependent boundary conditions and two supplementary transmission conditions at the point of discontinuity. We extend and generalize some approaches and results of the classic regular Sturm-Liouville problems to the similar problems with discontinuities. In particular, we introduce a special Hilbert space formulation in such a way that the problem under consideration can be interpreted as an eigenvalue problem for a suitable selfadjoint operator, construct the Green’s function and resolvent operator, and derive asymptotic formulas for eigenvalues and normalized eigenfunctions.Original Russian Text Copyright © 2005 Mukhtarov O. Sh. and Kadakal M.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 860–875, July–August, 2005.     

Energy and Foliations on Riemann Surfaces

We define the energy of foliations on Riemann surfaces. We prove that meromorphic vector fields are critical points and we compute their energies using the Green’s function. We then generalize the results to principal circle bundles over Riemann surfaces.Mathematics Subject Classification (2000): 53C12, 53C15.     

The Limit Shape of Young Diagrams for Multiplicative Statistics with Superpolynomial Growth

We obtain the limit shape of Young diagrams for a class of multiplicative statistics. In particular, we find the limit shape corresponding to J. Green’s function enumerating the characters of GL(n, F_q). Bibliography: 2 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 301, 2003, pp. 219–228.     

On the boundary value problem for the Sturm–Liouville equation with the discontinuous coefficient

We consider a boundary value problem for the Sturm–Liouville equation with piecewise‐constant leading coefficient. We prove that some integral representations for the solutions of the considered equation can be obtained by using classical transformation operators for the Sturm–Liouville operator at the end points of a finite interval. We also investigate the spectral characteristics of the boundary value problem, prove the completeness and expansion theorem. Copyright © 2012 John Wiley & Sons, Ltd.     

Exact master equations describing reduced dynamics of the Wigner function

Master equations of different types describe the evolution (reduced dynamics) of a subsystem of a larger system generated by the dynamic of the latter system. Since, in some cases, the (exact) master equations are relatively complicated, there exist numerous approximations for such equations, which are also called master equations. In the paper, we develop an exact master equation describing the reduced dynamics of the Wigner function for quantum systems obtained by a quantization of a Hamiltonian system with a quadratic Hamilton function. First, we consider an exact master equation for first integrals of ordinary differential equations in infinite-dimensional locally convex spaces. After this, we apply the results obtained to develop an exact master equation corresponding to a Liouville-type equation (which is the equation for first integrals of the (system of) Hamilton equation(s)); the latter master equation is called the master Liouville equation; it is a linear first-order differential equation with respect to a function of real variables taking values in a space of functions on the phase space. If the Hamilton equation generating the Liouville equation is linear, then the vector fields that define the first-order linear differential operators in the master Liouville equations are also linear, which in turn implies that for a Gaussian reference state the Fourier transform of a solution of the master Liouville equation also satisfies a linear differential equation. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 203–219, 2005.     

Distribution Functions in Quantum Mechanics and Wigner Functions

We formulate and solve the problem of finding a distribution function F(r,p,t) such that calculating statistical averages leads to the same local values of the number of particles, the momentum, and the energy as those in quantum mechanics. The method is based on the quantum mechanical definition of the probability density not limited by the number of particles in the system. The obtained distribution function coincides with the Wigner function only for spatially homogeneous systems. We obtain the chain of Bogoliubov equations, the Liouville equation for quantum distribution functions with an arbitrary number of particles in the system, the quantum kinetic equation with a self-consistent electromagnetic field, and the general expression for the dielectric permittivity tensor of the electron component of the plasma. In addition to the known physical effects that determine the dispersion of longitudinal and transverse waves in plasma, the latter tensor contains a contribution from the exchange Coulomb correlations significant for dense systems.     

Numerical study of one-dimensional Vlasov-Poisson equations for infinite homogeneous stellar systems

We consider the gravitational Vlasov-Poisson (VP), or the so-called collisionless Boltzmann-Poisson equations for the self-gravitating collisionless stellar systems. We compute the solutions using a high-order discontinuous Galerkin method for the Vlasov equation, and the classical representation by Green’s function for the Poisson equation in the one-dimensional setting. We study both the case of damping and Jeans instability depending on the wavenumbers, which are taken to be greater than or less than the Jeans wavenumber, respectively. The method is shown to be stable, accurate and conservative. We report for the first time the BGK modes for the gravitational VP system, and we study the behavior of solutions associated with these various wavenumbers.     

Recurrence and Transience Criteria for Symmetric Hunt Processes

In this paper, we will discuss recurrence, transience and other potential theoretic aspects based on symmetric regular Dirichlet space. We will first deal with Dirichlet space with the strong local property and give a recurrence criterion in terms of exhaustion function. This criterion shows that recurrence automatically provides us with an exhaustion function which is usable to verify a Liouville property on subharmonic functions. Secondly, a recurrence criterion and a transience criterion for a Nonlocal Dirichlet space will be presented. Those criteria can be applied to Albeverio–Karwowski"s random walks on p-adic number field. Lastly, we will prove the assertions which cover other potential theoretic aspect of p-adic number field such as Liouville property on harmonic functions.     

The Laplace and poisson equations in Schwarzschild's space-time

The method of separation of variables is used to solve the Laplace equation in Schwarzschild's space-time. The solutions are given explicitly in series form and in terms of Legendre functions. Green's function is determined and remarks are made on the solution of Poisson's equation for a point source.