On the Asymptotic Dynamics of a Quantum System Composed by Heavy and Light Particles

We consider a non-relativistic quantum system consisting of K heavy and N light particles in dimension three, where each heavy particle interacts with the light ones via a two-body potential α V. No interaction is assumed among particles of the same kind. Choosing an initial state in a product form and assuming α sufficiently small we characterize the asymptotic dynamics of the system in the limit of small mass ratio, with an explicit control of the error. In the case K = 1 the result is extended to arbitrary α. The proof relies on a perturbative analysis and exploits a generalized version of the standard dispersive estimates for the Schrödinger group. Exploiting the asymptotic formula, an application to the problem of the decoherence effect produced on a heavy particle by the interaction with the light ones is also outlined.     

The laplacian in regions with many small obstacles: Fluctuations around the limit operator

We consider the Laplacian _m in ³ (or in a bounded region of ³) with Dirichlet boundary conditions on the surfaces of some identical (small) neighborhoods ofm randomly distributed points, in the limit whenm goes to infinity and their linear size decreases as 1/m. We give here a stronger form of the result showing the convergence of the above operator to – C(x), whereC(x) is the limit density of electrostatic capacity of the obstacles. In particular results on the rate of convergence and on the fluctuations of _m around the limit operator are given.     

Derivation and classical limit of the mean-field equation for a quantum Coulomb system: Maxwell-Boltzmann statistics

The mean-field density matrix of a changed plasma of quantum particles with Maxwell-Boltzmann statistics in a confining external potential is obtained as a limit of theN-body canonical states for suitably scaled charges. Also, it is shown that the density profile of the quantum mean-field theory converges to the solution of the classical mean-field equation when the Planck's constant tends to zero.     

On the Aharonov–Bohm Hamiltonian

Using the theory of self-adjoint extensions, we construct all the possible Hamiltonians describing the nonrelativistic Aharonov–Bohm effect. In general, the resulting Hamiltonians are not rotationally invariant so that the angular momentum is not a constant of motion. Using an explicit formula for the resolvent, we describe the spectrum and compute the generalized eigenfunctions and the scattering amplitude.     

The Boltzmann Equation for a One-Dimensional Quantum Lorentz Gas

We study the macroscopic behavior of a quantum particle under the action of randomly distributed scatterers on the real line. Each scatterer generates a δ-potential. We prove that, in the low density limit, the Wigner function of the system converges to a probability distribution satisfying a classical linear Boltzmann equation, with a scattering cross section computed according to the Quantum Mechanical rules. Received: 2 April 1998 / Accepted: 12 February 1999     

Tedarenes A and B: structural and stereochemical analysis of two new strained cyclic diarylheptanoids from the marine sponge Tedania ignis

Ring strain causes planar chirality in tedarenes A and B, two cyclic diarylheptanoids isolated from the marine sponge Tedania ignis. In both molecules, the chiral plane is an olefinic system, which is very rare among natural products. In tedarene A (1), interconversion is too fast to allow isolation of the enantiomeric atropisomers but still slow enough to cause coalescence of some (1)H and (13)C NMR signals at room temperature. In tedarene B (2), which also shows stable central and axial chirality, the two planar diastereomers are in slow equilibrium. Tedarene B is the smallest natural product with central, axial, and planar chirality in the same simple molecule. The identification of planar chirality as the difference between its conformational isomers allowed the use of theoretical prediction of the CD spectrum to determine the absolute configuration of the stereogenic carbon C-9 as well as of the biphenyl chiral axis.     

Manadoperoxides, a new class of potent antitrypanosomal agents of marine origin

Chemical investigation of the marine sponge Plakortis cfr. lita afforded a library of endoperoxyketal polyketides, manadoperoxides B-K (3-5 and 7-13) and peroxyplakoric esters B(3) (6) and C (14). Eight of these metabolites are new compounds and some contain an unprecedented chlorine-bearing THF-type ring in the side chain. The library of endoperoxide derivatives was evaluated for in vitro activity against Trypanosoma brucei rhodesiense and Leishmania donovani. Some compounds, such as manadoperoxide B, exhibited ultrapotent trypanocidal activity (IC(50) = 3 ng mL(-1)) without cytotoxicity. Detailed examination of the antitrypanosomal activity data and comparison with those available in the literature for related dioxane derivatives enabled us to draw a series of structure-activity relationships. Interestingly, it appears that minor structural changes, such as a shift of the methyl group around the dioxane ring, can dramatically affect the antitrypanosomal activity. This information can be valuable to guide the design of optimized antitrypanosomal agents based on the dioxane scaffold.     

The NLS Equation in Dimension One with Spatially Concentrated Nonlinearities: the Pointlike Limit

In the present paper, we study the following scaled nonlinear Schrödinger equation (NLS) in one space dimension: $$ i\frac{\rm d}{{\rm d}t}\psi^{\varepsilon}(t)=-\Delta\psi^{\varepsilon}(t) +\frac{1}{\varepsilon}V\left(\frac{x}{\varepsilon} \right)|\psi^{\varepsilon}(t)|^{2\mu}\psi^{\varepsilon}(t)\quad \varepsilon > 0\,\quad V\in L^1(\mathbb{R},(1+|x|){\rm d}x) \cap L^\infty(\mathbb{R}).$$ This equation represents a nonlinear Schrödinger equation with a spatially concentrated nonlinearity. We show that in the limit \({\varepsilon\to 0}\) the weak (integral) dynamics converges in \({H^1(\mathbb{R})}\) to the weak dynamics of the NLS with point-concentrated nonlinearity: $$ i\frac{{\rm d}}{{\rm d}t} \psi(t) =H_{\alpha} \psi(t) .$$ where H _α is the Laplacian with the nonlinear boundary condition at the origin \({\psi'(t,0+)-\psi'(t,0-)=\alpha|\psi(t,0)|^{2\mu}\psi(t,0)}\) and \({\alpha=\int_{\mathbb{R}}V{\rm d}x}\) . The convergence occurs for every \({\mu\in \mathbb{R}^+}\) if V ≥  0 and for every  \({\mu\in (0,1)}\) otherwise. The same result holds true for a nonlinearity with an arbitrary number N of concentration points.     

Singularly Perturbed Hamiltonians of a Quantum Rayleigh Gas Defined as Quadratic Forms

We consider a class of singular, zero-range perturbations of the quantum Hamiltonian of a system consisting of a test particle and N harmonic oscillators (Rayleigh gas). Using the theory of quadratic forms we construct the self-adjoint and bounded from below perturbed Hamiltonian and we give representation formulas for the resolvent and the unitary group. The one-dimensional, two-dimensional and three-dimensional cases are discussed. Mathematics Subject Classifications (2000) 47A07, 47N50.Gianfausto DellAntonio: On leave from Dipartimento di Matematica, Università di Roma La Sapienza, Italy