Désiré Bollé

Witten index, axial anomaly, and Krein’s spectral shift function in supersymmetric quantum mechanics

A new method is presented to study supersymmetric quantum mechanics. Using relative scattering techniques, basic relations are derived between Krein’s spectral shift function, the Witten index, and the anomaly. The topological invariance of the spectral shift function is discussed. The power of this method is illustrated by treating various models and calculating explicitly the spectral shift function, the Witten index, and the anomaly. In particular, a complete treatment of the two‐dimensional magnetic field problem is given, without assuming that the magnetic flux is quantized.

Low-energy parameters in nonrelativistic scattering theory

Abstract Low-energy quantities like the scattering length and effective range parameter range parameter are direetly derived from the threshold behavior of the scattering amplitude for Schrodinger Hamiltonians with short-range and Coulomb-type interactions. The possibility of zero-energy resonances and zero-energy bound states of the underlying Hamiltonian is explicitly taken into account. No spherical symmetry of the interaction is assumed.

Stability properties of potts neural networks with biased patterns and low loading

The q-state Potts glass model of neural networks is extended to include biased patterns. For a finite number of such patterns, the existence and stability properties of the Mattis states and symmetric states are discussed in detail as a function of the bias. Analytic results are presented for all q at zero temperature. For finite temperatures numerical results are obtained for q=3 and two classes of representative bias parameters. A comparison is made with the Hopfield model.

SELF-CONTROL IN SPARSELY CODED NETWORKS

A complete self-control mechanism is proposed in the dynamics of neural networks through the introduction of a time-dependent threshold, determined in function of both the noise and the pattern activity in the network. Especially for sparsely coded models this mechanism is shown to considerably improve the storage capacity, the basins of attraction and the mutual information content of the network.

Scattering theory for one-dimensional systems with ∝ dx V(x) = 0

Abstract Low-energy scattering for Schrodinger operators of the type H= −Δ + V in L2(R) with ∫Rdx V(x) = 0 is considered. The possibility of zero-energy eigenstates of H is taken into account explicitly. In particular, a Laurent expansion for the transition operator and recursion relations for its coefficients are provided and the leading behaviour of the scattering operator is given in all possible cases.

Mutual information of sparsely coded associative memory with self-control and ternary neurons

The influence of a macroscopic time-dependent threshold on the retrieval dynamics of attractor associative memory models with ternary neurons ¿-1, 0, +1¿ is examined. If the threshold is chosen appropriately as a function of the cross-talk noise and of the activity of the memorized patterns in the model, adapting itself in the course of the time evolution, it guarantees an autonomous functioning of the model. Especially in the limit of sparse coding, it is found that this self-control mechanism considerably improves the quality of the fixed-point retrieval dynamics, in particular the storage capacity, the basins of attraction and the information content. The mutual information is shown to be the relevant parameter to study the retrieval quality of such sparsely coded models. Numerical results confirm these observations.

The cavity approach to parallel dynamics of Ising spins on a graph

We use the cavity method to study the parallel dynamics of disordered Ising models on a graph. In particular, we derive a set of recursive equations in single-site probabilities of paths propagating along the edges of the graph. These equations are analogous to the cavity equations for equilibrium models and are exact on a tree. On graphs with exclusively directed edges we find an exact expression for the stationary distribution. We present the phase diagrams for an Ising model on an asymmetric Bethe lattice and for a neural network with Hebbian interactions on an asymmetric scale-free graph. For graphs with a nonzero fraction of symmetric edges the equations can be solved for a finite number of time steps. Theoretical predictions are confirmed by simulations. Using a heuristic method the cavity equations are extended to a set of equations that determine the marginals of the stationary distribution of Ising models on graphs with a nonzero fraction of symmetric edges. The results from this method are discussed and compared with simulations.

Time delay in N‐body scattering

We extend the theory of time delay to N‐body scattering. The known results relating time delay to the S matrix in the two‐body and three‐body problem suggest that these relationships are universal. Within the context of two‐Hilbert space N‐body scattering theory an abstract definition of time delay is provided. For all scattering processes initiated by the collision of two clusters a simple proof is constructed establishing the connection of time delay to the on‐shell S matrix and its energy derivatives. The definition of time delay and method of proof given here are compared with earlier approaches used in the three‐body problem.

On the overlap dynamics of multi-state neural networks with a finite number of patterns

Neural networks with multi-state neurons are studied in the case of low loading. For symmetric couplings satisfying a certain positivity condition, a Lyapunov function is shown to exist in the space of overlaps between the instantaneous microscopic state of the system and the learned patterns. Furthermore, an algorithm is derived for zero temperature to determine all the fixed points. As an illustration, the three-state model is worked out explicitly for Hebbian couplings. For finite temperature the time evolution of the overlap is studied for couplings which need not be symmetric. The stability properties are discussed in detail for the three-state model. For asymmetric couplings limit-cycle behaviour is shown to be possible.

On the parallel dynamics of the Q-state potts and Q-ising neural networks

Using a probabilistic approach, the parallel dynamics of theQ-state Potts andQ-Ising neural networks are studied at zero and at nonzero temperatures. Evolution equations are derived for the first time step and arbitraryQ. These formulas constitute recursion relations for the exact parallel dynamics of the extremely diluted asymmetric versions of these networks. An explicit analysis, including dynamical capacity-temperature diagrams and the temperature dependence of the overlap, is carried out forQ=3. Both types of models are compared.