Effective potential, Bohm’s potential plus classical potential, analysis of quantum transmission

The influence of spreading, in wavepacket transmission across a potential barrier has been analyzed, by considering several collisions between a wavepacket and a potential barrier, in which the initial distance between the packet and the barrier—the launching distance, is changed. An effective total potential (Bohm’s quantum potential plus classical potential), has been used, to show that, for suitable classical barrier widths and heights, light masses, as well as mean collision energies, one should expect an increase of the quantum transmission factor as the initial wavepacket—barrier distance is decreased. Numerically converged time-dependent wavepacket propagation calculations confirm that trend, leading to an increase as high as 20% per ?, in thin square and Eckart barrier problems. Possibilities of experimentally measuring this effect are also analyzed.     

Family algebras and generalized exponents for polyvector representations of simple Lie algebras of type <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

We give an explicit formula for the exterior powers ∧ ^k π ₁ of the defining representation π ₁ of the simple Lie algebra ?ο(2n + 1, ?). We use the technique of family algebras. All representations in question are children of the spinor representation σ of g2ο(2n + 1, ?). We also give a survey of main results on family algebras.     

Complex multiplication and parity in Iwasawa theory

We give a new, somewhat elementary method for proving parity results about Iwasawa-theoretic Selmer groups and apply our method to certain Galois representations which are not self-dual. The main result is essentially that Iwasawa's λ-invariants for these representations over dihedral -extensions are even. Our approach is a specialization argument and does not make use of Neková?'s deformation-theoretic Cassels pairing, though Neková?'s theory implies our results. Examples of the representations we consider arise naturally in the study of CM abelian varieties defined over the totally real subfield of the reflex field of the CM type. We also discuss connections with “large Selmer rank” in the sense of Mazur-Rubin and give several examples in the context of abelian varieties and modular forms.     

Multiplicative representation of bilinear operators

We establish that each lattice bimorphism from the Cartesian product of two vector lattices into a universally complete vector lattice is representable as the product of two lattice homomorphisms defined on the factors. This fact makes it possible to reduce the problem to the linear case and obtain some results on representation of an order bounded disjointness preserving bilinear operator as a strongly disjoint sum of weighted shift or multiplicative operators.     

Probabilistic Representations of Solutions of the Forward Equations

In this paper we prove a stochastic representation for solutions of the evolution equation

Sobolev-type integral representations for functions defined on Carnot groups

We obtain some integral representations of the form f(x) = P(f) + K(?f) on the Carnot groups, where P(f) is a polynomial and K is an integral operator with a specific singularity. These representations are employed to prove the weak Poincaré inequality.     

Remarks on Naimark's duality

We present an extension of a version of Naimark's dilation theorem which states that complete systems in a Hilbert space are projections of -linearly independent systems of elements of an ambient Hilbert space. This result is presented in the context of other known extensions of Naimark's theorem.

     

Maximum of the modulus of kernels in Gauss-Turán quadratures

We study the kernels in the remainder terms of the Gauss-Turán quadrature formulae for analytic functions on elliptical contours with foci at , when the weight is a generalized Chebyshev weight function. For the generalized Chebyshev weight of the first (third) kind, it is shown that the modulus of the kernel attains its maximum on the real axis (positive real semi-axis) for each . It was stated as a conjecture in [Math. Comp. 72 (2003), 1855-1872]. For the generalized Chebyshev weight of the second kind, in the case when the number of the nodes in the corresponding Gauss-Turán quadrature formula is even, it is shown that the modulus of the kernel attains its maximum on the imaginary axis for each . Numerical examples are included.

     

Explicit rational solutions of Knizhnik-Zamolodchikov equation

We consider the Knizhnik-Zamolodchikov system of linear differential equations. The coefficients of this system are rational functions generated by elements of the symmetric group _n . We assume that parameter ρ = ±1. In previous paper [5] we proved that the fundamental solution of the corresponding KZ-equation is rational. Now we construct this solution in the explicit form.