The Flux-Across-Surfaces Theorem and Zero-Energy Resonances

In this contribution we shall first introduce the Flux Across Surfaces (FAS) theorem, placing it in the general context of the Quantum Scattering Theory. Then we shall review briefly the theory of resonances in non-relativistic Quantum Mechanics and outline a proof of the FAS theorem for non-relativistic potential scattering, which covers also the case in which there is a zero energy resonance.     

Selfduality and topological-like properties of lattice gauge field theories. A proposal

We introduce for lattice gauge theories an analogue of the Pontrjagin index and a notion of selfduality and antiselfduality. Selfdual and antiselfdual configurations on the lattice have much of the same properties (with some remarkable differences) as the corresponding configurations on the continuum, to which they converge when the lattice spacing goes to zero.On leave of absence from Istituto di Fisica dell'Università di Milano, Via Celoria 16, 20133 Milano, Italy.Research partially supported by Consiglio Nazionale delle Ricerche and Fondazione A. Della Riccia     

Ionization for Three Dimensional Time-Dependent Point Interactions

We study the time evolution of a three dimensional quantum particle under the action of a time-dependent point interaction fixed at the origin. We assume that the strength of the interaction (t) is a periodic function with an arbitrary mean. Under very weak conditions on the Fourier coefficients of (t), we prove that there is complete ionization as t, starting from a bound state at time t=0. Moreover we prove also that, under the same conditions, all the states of the system are scattering states.On leave from Dipartimento di Matematica, Università di Roma, La Sapienza, Italy.     

Convergence of Nelson fiffusions

Let _t, _t ⁿ ,n1, be solutions of Schrödinger equations with potentials form-bounded by –1/2 and initial data inH ¹( ^d ). LetP, P ⁿ ,n1, be the probability measures on the path space =C(₊, ^d ) given by the corresponding Nelson diffusions. We show that if { _t ⁿ } _n1 converges to _t inH ¹( ^d ), uniformly int over compact intervals, then converges to in total variation t0. Moreover, if the potentials are in the Kato classK _d , we show that the above result follows fromH ¹-convergence of initial data, andK _d -convergence of potentials.     

On some groups of automorphisms of physical observables

Given a weakly continuous one-parameter group of automorphisms of aC*-algebra of operators on a Hilbert space we show that it is implementable by a strongly continuous one-parameter group of unitary operators belonging to the weak closure of , provided that a certain condition — akin to the boundedness from below of the spectrum of the generators — is satisfied.On leave from the Istituto di Fisica Teorica, Universitá di Napoli.     

Particles and “bumps” in quantum field configurations

We study an approach to making a precise identification of particles with bumps in quantum field configurations. The problem requires some effort because typical field configurations are distributions and not elements of the classical configuration space at all. We show that the part of a configuration consisting of vacuum fluctuations may be identified and filtered out in a very natural way, leaving a function which is in the classical configuration space. The filtered field configuration depends on the state of the field, and for ann-particle state withn particles well localized and well separated in the Newton-Wigner sense, the filtered field configuration hasn bumps located where the particles are. We close with a discussion of observation in the Schrödinger representation for the free field in terms of our results.This paper is dedicated to the memory of Paola Calderoni.     

Dynamics on quantum graphs as constrained systems

We study the possibility of regarding the dynamics on a quantum graph as limit, as a small parameter ∈ → O, of a dynamics with a strong confining potential. We define a projection operator along the first eigenfunction of a transversal operator and, under suitable assumptions, we prove that the projection of the solution strongly converges along subsequences to a function that satisfies the Schrödinger equation on each open edge of the graph. Moreover the limit dynamics is unitary. If the limit is independent of the subsequence, one has a limit one-parameter group, generated by one of the self-adjoint extensions of a symmetric operator defined on the open graph (with the vertices deleted). The crucial role of the shape of the confining potential at the vertices is pointed out.     

Structure of the algebras of some free systems

We give a detailed analysis of the structure of some Von Neumann algebras which describe free relativistic fields or infinite systems of free particles with finite density.