Two Cold Atoms in a Harmonic Trap

Two ultracold atoms moving in a trap interact weakly at a very short distance. This interaction can be modeled by a properly regularized contact potential. We solve the corresponding time-independent Schrödinger equation under the assumption of a parabolic, spherically symmetric trapping potential.     

The Ornstein-Uhlenbeck semigroup in an infinite dimensional L2 setting

We study the number operator, N, of quantum field theory as a partial differential operator in infinitely many variables. Informally Nu(x) = ?Δu(x) + x · grad u(x). A large core for N is constructed which is invariant under e^?tN and on which this informal expression may be given a precise and natural meaning.     

Some regularity properties of diffusion processes on abstract Wiener space

Diffusion processes on an abstract Wiener space are constructed from fundamental solutions of second-order parabolic equations with variable coefficients. The transition probabilities of such processes are compared with those of the Wiener process, and continuity of sample paths is established. Several operator semigroups associated with the processes are defined (one locally), and some regularity properties of these semigroups are established.     

Quadratic forms over formally real fields with eight square classes

Complete classification of formally real fields with 8 square classes with respect to the behaviour of quadratic forms is given. Two fields F and K are equivalent with respect to quadratic forms if the quadratic form schemes of the two fields are isomorphic or in other words, if the Witt rings W(F) and W(K) are isomorphic. It is shown here that for formally real fields with 8 square classes there are exactly seven possible quadratic form schemes and for each of the seven schemes a formally real field with 8 square classes and the given scheme is constructed.     

Particle representations of canonical commutation relations

The aim of the present paper is to find a necessary and sufficient condition for the existence of the operator of the total number of particles in a representation of canonical commutation relations. The result is formulated by means of generating functionals of representations. It is shown that an irreducible representation possesses a (generalized) number operator if and only if the representation obtained by averaging its generating functional over the group of phase transformations is a factor representation of type I.     

Analytical evaluation of lower critical hydrostatic pressure for imperfect cylindrical shells of moderate length

The paper applies generalized power series to determine effectively the lower critical hydrostatic pressure for imperfect cylindrical shells. The basic equations were assumed in the form derived by W.A. Nash. Two variants as regards the number of circumferential half-wavelengths are discussed: m constant during buckling, thus corresponding to upper critical pressure, and m furnishing minimal value of lower critical pressure. It is found that the dependence of lower critical pressure on an imperfection parameter V is linear, i.e. smaller than in the case of the upper critical pressure which decreases with $\text{V}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}$.     

Direct Synthesis of Amphiphilic α-, β-, and γ-Cyclodextrins

ABSTRACT

The clean one step synthesis of the amphiphilic α-, β-, and γ-cyclodextrins starting from per-(6-bromo-6-deoxy)-α-, -β-, and -γ-cyclodextrins is described. The role of the lipophilic tail is played by various aryl groups (phenyl, p-bromophenyl, p-O-butoxyphenyl, p-pentylphenyl, and o-, m-, and p-nitrophenyl) linked by a thioether bridge to the position C-6 of each glucopyranose unit. The yields of the S-alkylation reactions were very high (85-95%).