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.     

On the Zeros of Polynomials Satisfying Certain Linear Second-Order ODEs Featuring Many Free Parameters

Certain techniques to obtain properties of the zeros of polynomials satisfying second-order ODEs are reviewed. The application of these techniques to the classical polynomials yields formulas which were already known; new are instead the formulas for the zeros of the (recently identified, and rather explicitly known) polynomials satisfying a (recently identified) second-order ODE which features many free parameters and only polynomial solutions. Some of these formulas have a Diophantine connotation. Techniques to manufacture infinite sequences of second-order ODEs featuring only polynomial solutions are also reported.     

The Z<Superscript>-1</Superscript> expansions of the total non-relativistic energies for three- and four-electron systems

Analytical formulas are derived for the Z^-1 (Q^-1) expansions of the total energies in the two-, three- and four-electron atomic systems. The coefficients in these formulas are determined from the results of accurate and highly accurate computations with correlated wave functions. The accuracy of our formulas is sufficient to evaluate the total ground state energy of an arbitrary non-relativistic ion with two, three and four electrons.     

Changes in the spectral degree of polarization of a partially coherent dark hollow beam in the turbulent atmosphere for on-axis and off-axis propagation point

We report analytic formulas for the elements of the 2 × 2 cross spectral density matrix of a partially coherent dark hollow beam (DHB) with circular symmetry propagating through the turbulent atmosphere. In this paper, from these formulas changes in the spectral degree of polarization of partially coherent DHB for on-axis and off-axis propagation point and for near and far fields are determined for different source conditions.     

Formulas for the characteristic polynomial coefficients of the pendant graphs of linear chains,cycles and stars

Formulas for the characteristic polynomial (CP) coefficients of three classes of (n + p)-vertex graphs, i.e. linear chains, cycles and stars where p pendant vertices are attached to n base vertices in one-to-one correspondence (p = 0, 1, 2, …, n), have been developed. Such pendant graphs become reciprocal graphs for linear chains and cycles if p = n. The n-vertex star graphs follow the same rule as paths and cycles, they become reciprocal on adding a pendant vertex to each of n vertices. The formulas so developed have been expressed in matrix product and in analytical forms for the three classes of graphs that require only the values of n and p for calculation of the respective CP coefficients. Such formulas have the general applicability for a large variety of molecular graphs with varying n and p and have been shown to be reduced to the corresponding formulas for reciprocal graphs that are the special cases of the graphs discussed here.     

On the Renormalized Volume of Hyperbolic 3-Manifolds

The renormalized volume of hyperbolic manifolds is a quantity motivated by the AdS/CFT correspondence of string theory and computed via a certain regularization procedure. The main aim of the present paper is to elucidate its geometrical meaning. We use another regularization procedure based on surfaces equidistant to a given convex surface ?N. The renormalized volume computed via this procedure is equal to what we call the W-volume of the convex region N given by the usual volume of N minus the quarter of the integral of the mean curvature over ?N. The W-volume satisfies some remarkable properties. First, this quantity is self-dual in the sense explained in the paper. Second, it verifies some simple variational formulas analogous to the classical geometrical Schläfli identities. These variational formulas are invariant under a certain transformation that replaces the data at ?N by those at infinity of M. We use the variational formulas in terms of the data at infinity to give a simple geometrical proof of results of Takhtajan et al on the Kähler potential on various moduli spaces.     

An explicit *-product on the cotangent bundle of a Lie group

We give explicit formulas for a ^*-product on the cotangent bundle T ^* G of a Lie group G; these formulas involve on the one hand the multiplicative structure of the universal enveloping algebra U(G) of the Lie algebra G of G and on the other hand bidifferential operators analogous to the ones used by Moyal to define a ^*-product on IR²ⁿ.Chargé de recherches au FNRS, on leave of absence from Université libre de Bruxelles.     

On the q-analog of the Laplace transform

In this paper, we consider the q-analog of the Laplace transform and investigate some properties of the q-Laplace transform. In our investigation, we derive some interesting formulas related to the q-Laplace transform.     

Integral Formulas for the Asymmetric Simple Exclusion Process

In this paper we obtain general integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer lattice with nearest neighbor hopping rates p to the right and q = 1−p to the left. For the most part we consider an N-particle system but for certain of these formulas we can take the limit. First we obtain, for the N-particle system, a formula for the probability of a configuration at time t, given the initial configuration. For this we use Bethe Ansatz ideas to solve the master equation, extending a result of Schütz for the case N = 2. The main results of the paper, derived from this, are integral formulas for the probability, for given initial configuration, that the m ^th left-most particle is at x at time t. In one of these formulas we can take the limit, and it gives the probability for an infinite system where the initial configuration is bounded on one side. For the special case of the totally asymmetric simple exclusion process (TASEP) our formulas reduce to the known ones.     

Coulomb break-up of the deuteron by the positive muon

The μ⁺d → μ⁺pn process is described in the framework of three-body scattering theory which includes two charged particles. Explicit formulas for the break-up amplitude are given, and muon spectra are calculated in a simple approximation. n-p off-energy-shell effects are investigated.     

On the Determination of the Absolute Intensities of Vibrational-Rotational Lines of XY3 Molecules with C 3v Symmetry

Analytical formulas are derived that describe the dependences of the transition probabilities in vibrational-rotational spectra of XY₃-type molecules with C _3v symmetry on the rotational quantum numbers.     

Formulas for the derivative and critical points of topological entropy for Anosov and geodesic flows

This paper represents part of a program to understand the behavior of topological entropy for Anosov and geodesic flows. In this paper, we have two goals. First we obtain some regularity results forC ¹ perturbations. Second, and more importantly, we obtain explicit formulas for the derivative of topological entropy. These formulas allow us to characterize the critical points of topological entropy on the space of negatively curved metrics.Partially supported by NSF grant DMS-8514630Chaim Weizmann Research Fellow and NSF postdoctoral Research Fellow     

Analytical representation of broadening coefficients of vibrational-rotational absorption lines of linear molecules

Analytical formulas are proposed for the coefficients that determine broadening of vibrational-rotational absorption lines of diatomic and linear molecules in the case when the broadening is caused by the pressure of buffer gases or linear molecules. The formulas are obtained by the parameterization of the semiclassical method relations with allowance for the predominant contribution related to polarization or dipole-quadrupole interaction. The parameters of the analytical formulas are determined for different vibrational bands of HCl and HF molecules in the case of argon, nitrogen, and air broadening, as well as for CO and CO₂ molecules in the case of broadening by collisions with CO, CO₂, N₂, O₂, and H₂O.     

Influence of the separation energy on the radius of neutron rich nuclei

The intensive studies of equilibration processes in heavy ion reactions have produced a need for information on nuclear level densities at high energies. In a recent paper, it was concluded that standard Fermi gas formulas will be incorrect by exponential factors at energies above 100 MeV. Exact calculations of the nuclear level density in bases as large as 10³⁸ have been made and are compared with Fermi gas formulas. Two possible alternative forms are considered. Both forms produce much better agreement at high energies than does the Fermi gas model. All calculations reported are for non-interacting Fermions, but the effects expected from the two-body interaction are briefly examined. These considerations have consequences not only in heavy ion physics but also in astrophysics.     

Reflection and transmission coefficients of electromagnetic waves with different polarizations on a one-dimensional layer of an inhomogeneity of the medium

We have obtained strict formulas (without approximations) for the reflection and transmission coefficients of electromagnetic waves with an arbitrary polarization in an isotropic medium with a one-dimensional layer of an inhomogeneity of an arbitrary profile playing the role of a transition layer between two different homogeneous media. Since these formulas involve values that are found by numerically integrating the first-order ordinary differential equations, the result of this paper is a fast and convenient method for calculating the above coefficients. The calculated results for two particular cases are given. An analytical limiting passage from the obtained formulas to the particular case of a sharp interface between the media (i.e., to known Fresnel formulas) is presented.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 39, No. 9, pp. 1075–1086, September, 1996.This work was supported by the Russian Foundation for Fundamental Research.     

Coulomb effects in relativistic nuclear collisions

We derive simple analytical formulas for Coulomb final-state interactions and apply them to the analysis of recent data on nuclear collisions. The π^?/π⁺ ratio, the π⁺ inclusive cross section, and the n/p ratio are studied. A relativistic field theoretic model is used to derive the formulas to first order in Zα. Using well-known non-perturbative results, we recast those formulas into an approximate non-perturbative form valid when finite-size effects are negligible. This allows us to calculate the important k → 0 limit. The final formulas are covariant and take into account multiple independently moving charged fragments of finite size and finite thermal expansion velocities. Our studies demonstrate analytically the complexity and importance of Coulomb distortions in nuclear collisions.     

On the “Matrix Approach” to Interacting Particle Systems

Derrida et al. and Schütz and Stinchcombe gave algebraic formulas for the correlation functions of the partially asymmetric simple exclusion process. Here we give a fairly general recipe of how to get these formulas and extend them to the whole time evolution (starting from the generator of the process), for a certain class of interacting systems. We then analyze the algebraic relations obtained to show that the matrix approach does not work with some models such as the voter and the contact processes.     

Analytical formulas for complex permittivity of periodic composites. Estimation of gain and loss enhancement in active and passive composites

ABSTRACT

The asymptotic homogenization method is applied to complex dielectric periodic composites. An equivalence to coupled dielectric problems with real coefficients is shown. This is similar to a piezoelectric problem: an out-plane mechanical displacement and an in-plane electric potential establishing a correspondence principle. Closed-form formulas for the complex dielectric effective tensor in the case of a square array of circular inclusions embedded in a matrix are given. These formulas are written in terms of a real and symmetric matrix which facilitates the implementation of the computational scheme. We also get similar formulas for multilayered complex dielectric composites. The real closed-form formulas are advantageous for estimating gain and loss enhancement properties of active and passive composites in certain volume fraction intervals. Numerical computations are performed and the results are compared with other approaches showing the usefulness of the obtained formulas. This may be of interest in the context of metamaterials.     

Vicious random walkers in the limit of a large number of walkers

The vicious random walker problem on a line is studied in the limit of a large number of walkers. The multidimensional integral representing the probability that thep walkers will survive a timet (denotedP _t ^(p) ) is shown to be analogous to the partition function of a particular one-component Coulomb gas. By assuming the existence of the thermodynamic limit for the Coulomb gas, one can deduce asymptotic formulas forP _t ^(p) in the large-p, large-t limit. A straightforward analysis gives rigorous asymptotic formulas for the probability that after a timet the walkers are in their initial configuration (this event is termed a reunion). Consequently, asymptotic formulas for the conditional probability of a reunion, given that all walkers survive, are derived. Also, an asymptotic formula for the conditional probability density that any walker will arrive at a particular point in timet, given that allp walkers survive, is calculated in the limittp.