The cauchy problem for the Landau-Lifschitz equation on a finite interval

The Landau-Lifshitz equation for ferromagnets is reduced to a matrix Riemann boundary problem on a torus. This problem has been solved effectively. The time-dependence is re-established by a nonlinear equation in the function of a single variable.     

Solution of Fokker-Planck equation using Trotter's formula

Solution of Fokker-Planck equation using Trotter's formula is discussed. The method is illustrated on the linear Fokker-Planck equation and the Ornstein-Uhlenbeck solution is obtained. For the case of a general nonlinear Fokker-Planck equation the method yields an integral representation amenable to approximations. In the lowest order approximation Suzuki's scaling result emerges. Physical interpretation and limitations of the approximations are also discussed.     

A new look at the harmonic oscillator problem in a finite-dimensional Hilbert space

In this Letter some basic properties of a truncated oscillator are studied. By using finite-dimensional representation matrices of the truncated oscillator we construct new parasupersymmetric schemes and remark on their relevance to the transition operators of the non-interacting N-level system endowed with bosonic modes.     

A Hilbert space approach to the Lippmann-Schwinger equation in momentum representation

We study the Lippmann-Schwinger equation for the quantum mechanical two-body scattering problem. We propose a Hilbert space approach in momentum-angular momentum representation. Imposing a Hölder integrability condition on the potential, the kernel of the integral equation becomes a compact operator in an adequate Hilbert space H⁰. We show that expansion into orthogonal polynomials becomes very simple, and we give an application to the three-particle problem.     

Exact solution to the cauchy problem for a generalized “linear” vectorial Fokker-Planck equation: Algebraic approach

The exact solution to the Cauchy problem for a generalized “linear” vectorial Fokker-Planck equation is found by using the disentangling techniques of Feynman and algebraic (operational) methods.     

Solution of the Fokker-Planck equation for the shock wave problem

An eigenexpansion solution of the time-independent Brownian motion Fokker-Planck equation is given for a situation in which the external acceleration is a step function. The solution describes the heavy-species velocity distribution function in a binary mixture undergoing a shock wave, in the limit of high dilution of the heavy species and negligible width of the light-gas internal shock. The diffusion solution is part of the eigenexpansion. The coefficients of the series of eigenfunctions are obtained analytically with transcendentally small errors of order exp(–1/M), whereM 1 is the mass ratio. Comparison is made with results from a hypersonic approximation.     

A generalized Schrödinger formalism as a Hilbert space representation of a generalized Liouville equation

A generalized Schrödinger formalism has been presented which is obtained as a Hilbert space representation of a Liouville equation generalized to include the action as a dynamical variable, in addition to the positions and the momenta. This formalism applied to a classical mechanical system had been shown to yield a similar set of Schrödinger like equations for the classical dynamical system of charged particles in a magnetic field. The novel quantum-like predictions for this classical mechanical system have been experimentally demonstrated and the results are presented.     

Discussion of the Królikowski-Rzewuski equation for projection of a state-vector onto one-dimensional subspace of Hilbert space

A detailed discussion is given of the quasipotential governing time evolution of a projection of state-vector onto a definite direction in the Hilbert space of states. An error is pointed out in the general formula for the quasipotential given by Królikowski and Rzewuski, and the correct formula is presented. An estimation is made of the quality of approximation consisting in replacement in the equation of evolution of a nonstationary Hamiltonian by its stationary limiting value.     

A method of cauchy integral equation for non-coherent transfer in half-space

A technique is presented which allows easy construction of solutions for various half-space problems arising in non-coherent radiative transfer with complete redistribution. By use of an inverse Laplace transform method, Wiener-Hopf integral equations are reduced to Cauchy-type singular integral equations. The factorization technique used by Case and Zweifel for coherent scattering can then be carried over to non-coherent transfer. The method is applied to the inhomogeneous integral equation for the source function of a two-level atom, previously solved by Ivanov. It is also applied to the conservative, homogeneous case and to singular Wiener-Hopf equations arising from asymptotic expansions in the limit of vanishing probability of collisional destruction ?. Consequences for the scaling laws in a finite slab are examined in a companion paper.     

Solutions for a non-Markovian diffusion equation

Solutions for a non-Markovian diffusion equation are investigated. For this equation, we consider a spatial and time dependent diffusion coefficient and the presence of an absorbent term. The solutions exhibit an anomalous behavior which may be related to the solutions of fractional diffusion equations and anomalous diffusion.     

Solution of the initial value problem for the sine-Gordon equation using a Kac-Moody algebra

We solve the classical sine-Gordon equation using a Lax pair belonging to a Kac-Moody algebra. By realising the algebra in terms of fermionic currents we reduce the initial value problem to the evaluation of a fermionic propagator as a sum of Feynman diagrams.     

A general formula for analytic reduction of multi-loop tensor Feynman integrals

We prove a general formula for analytic reduction of tensor integrals which appear in calculations of multi-loop Feynman diagrams in quantum field theory models.     

Solution of the one-dimensional diffusion equation for the hadronic cosmic-ray component in the atmosphere

Summary We solve the one-dimensional diffusion equation for the hadronic cosmic-ray component in the atmosphere assuming for the interaction mean free path a ln² E-dependence on the energy and comparing with mountain experimental data. The authors of this paper have agreed to not receive the proofs for correction. Partial financial support of FAPESP and CNPq, Brasil.     

有限维希尔伯特空间中的非线性相干态

在有限维希尔伯特空间中构造了非线性相干态,并研究了该量子态的压缩效应和反聚束效应,给出了压缩效应和反聚束效应与有限维希尔伯特空间维数关系的数值计算结果.     

Feynman formulas for the diffusion of particles with position-dependent mass

In the present paper, Feynman formulas are obtained for Schrödinger semigroups generated by self-adjoint operators which are perturbations of self-adjoint extensions of the second-order Hamiltonian operator ?Δ _g,0/2+V (throughout the paper, the coefficient ?1/2 at Δ _g,0 is omitted to simplify the formulas) which describe the diffusion of a quasiparticle with position-dependent mass varying jump-like on a line. Every extension of this kind is defined by some invertible operator and is characterized by matching conditions at a jump point. The Schrödinger semigroups generated by self-adjoint Laplace operators and defined by the corresponding boundary conditions define solutions of initial-boundary value problems. In turn, the term “Feynman formulas” is applied (in the present case) to an explicit representation of the Schrödinger semigroup \(e^{t\hat H^T } \) in the form of a limit of integrals of finite multiplicity over Cartesian powers of some configuration space. In essence, the Feynman-Kac formula is a “probabilistic interpretation” of the Feynman formulas. Namely, the multiple integrals in the Feynman formulas approximate integrals against some measures on the space of trajectories (functions defined on an interval of the real line and ranging in the configuration space). Thus, the Feynman formulas enable one to evaluate integrals over spaces of trajectories. A crucial role in the proof of the Feynman formulas is played by the Chernoff theorem, which is a generalization of the famous Trotter formula. The result proved in the present paper is a demonstration of a part of the results recently announced by O. G. Smolyanov and H. von Weizäcker (“Feynman Formulas Generated by Self-Adjoint Extensions of the Laplacian,” Dokl. Ross. Akad. Nauk 426 (2), 162–165 (2009) [Doklady Mathematics, 2009 79 (3), 335–338 (2009)]). The formulations of the results in question are inessentially modified here.     

The first initial-boundary value problem for the space charge stratification equation in semiconductors

The internal and external initial-boundary value problems for 3D equation ?/?t (Δu ? u) ? u = 0 are considered. The problems are posed under the Dirichlet boundary condition on a Lyapunov surface. Using the method of dynamic potentials, the theorems of the unique existence of the solution are proved.