Disappearance of time integrals of exact memory functions in time-convolution generalized master equations

Memory functions in time-convolution Generalized Master Equations (GME) for probabilities of finding a general system (interacting by a general coupling with a true thermodynamic bath) in individual states are considered without resorting to any approximation. After taking the thermodynamic bath limit, time integrals from zero to infinite times of the memories are considered. It is argued that these integrals entering, e.g., the usual naive Markov approximation converting GME the Pauli master (PME) equations are exactly zero. This implies long-time tails of memories (unobtainable by perturbational expansions) and slower-than-exponential long-time asymptotics of relaxation.     

Three-particle integrals with Bessel functions

Analytical formulas for some useful three-particle integrals are derived. Many of these integrals include Bessel and/or trigonometric functions of one and two interparticle (relative) coordinates r ₃₂, r ₃₁, and r ₂₁. The formulas obtained in such an analysis allow us to consider three-particle integrals of more complicated functions of relative/perimetric coordinates. In many actual problems such three-particle integrals can be found in matrix elements of the Hamiltonian and other operators.     

Bohr-Sommerfeld conditions in geometric quantization

The corrected Bohr-Sommerfeld quantum conditions, ∫ pdq?d = integer, are studied in the framework of geometric quantization. It is shown, in the representation given by a polarization F, that a half-form corresponds to a wave function only if it vanishes on all closed curves with tangent vectors in F for which the quantum condition is not satisfied. The constant d is determined, for each closed curve y, by the element of the holonomy group of a bundle of metalinear frames for F induced by y. This result is applied to a one-dimensional harmonic oscillator and a two-dimensional relativistic Kepler problem. In the case of the one-dimensional harmonic oscillator there are two possibilities of choosing a metalinear frame bundle for F. One choice leads to the original Bohr-Sommerfeld condition while the other leads to the corrected version with d = $\text{1}\text{2}$. Similarly, choosing different metalinear frame bundles for F, we get for the relativistic Kepler problem the fine structures of the energy levels corresponding to spin 0 and spin $\text{1}\text{2}$.     

Calculation of integrals involving characteristic beam functions

An analytical procedure is presented for evaluating some important integrals involving characteristic beam functions. These integrals were encountered in the vibration analyses of constrained and unconstrained circular cylindrical shells. The analysis will also be useful in many other problems where characteristic beam functions are involved and will enable complicated and computer-time-consuming numerical integration to be avoided.     

Bohr-Sommerfeld quantization of spin Hamiltonians

The Bohr-Sommerfeld rule for a spin system is obtained, including the first quantum corrections. The rule applies to both integer and half-integer spin. It is tested for various models, in particular, the Lipkin-Meshkov-Glick model, and found to agree very well with exact results.     

Equivariant Asymptotics for Bohr-Sommerfeld Lagrangian Submanifolds

Suppose given a complex projective manifold M with a fixed Hodge form Ω. The Bohr-Sommerfeld Lagrangian submanifolds of (M,Ω) are the geometric counterpart to semi-classical physical states, and their geometric quantization has been extensively studied. Here we revisit this theory in the equivariant context, in the presence of a compatible (Hamiltonian) action of a connected compact Lie group.     

WKB and exact wave functions for inverse power-law potentials

We compare WKB and exact wave functions for inverse power-law potentials and derive a universal algebraic formula reproducing reflection phases and scattering phase shifts with uniform high accuracy for any wave length. Appropriately modified quantization gives competitive accuracy for the spiked harmonic oscillator. The asymptotic accuracy of WKB wave functions is discussed.     

Vertex functions from the Mandelstam string path integrals

As a pedagogical exercise we derive the vertex functions in the light-cone gauge for an arbitrary string state by direct integration of the Mandelstam path integral formula.     

Functional integrals for the partition functions of dual strings

We show how the partition functions and helicity partition functions of the known dual strings can be obtained from functional integrals. We define these integrals using a cut-off which is essentially equivalent to Ramanujan's definition of divergent sums. This leads to a very compact algorithm.     

计算含三角函数无穷积分的新方法

在留数定理及Jordan引理的基础上,提出了一个新的引理,进而发展了计算含三角函数无穷积分的一种新方法.作为应用,计算了几个典型的含三角函数的无穷积分.     

Bohr-Sommerfeld quantization condition for the Gross-Pitaevskii equation

The discrete spectrum of the nonlinear eigenvalue problem associated to the one-dimensional Gross-Pitaevskii equation with a smooth potential is studied in the quasiclassical limit. We particularly focus on the corrections to the Bohr-Sommerfeld quantization rule for the excited energy levels due to the nonlinearity. Explicit predictions are obtained analytically for these corrections and are supported by numerical computations.     

An exact treatment of some integrals appearing in the three-body contributions to transport coefficients

The convergence properties of the integral corresponding to the three-body cyclic collisions in the two-dimensional density expansion of transport coefficients are investigated. It is shown exactly that the integral considered by Oppenheim et al. and by Eu diverges logarithmically in the switching parameter ε.     

On exact singular wave functions of two identical particles

We discuss the occurrence and properties of exact singular anyonic wave functions describing stationary states of two identical charged particles moving on a plane and under the influence of a perpendicular uniform magnetic field. Received 25 September 2001 / Received in final form 4 December 2001 Published online 2 October 2002 RID="a" ID="a"e-mail: truong@u-cergy.fr